Bit Error Rate (BER) for BPSK modulation

In this post, we will derive the theoretical equation for bit error rate (BER) with Binary Phase Shift Keying (BPSK) modulation scheme in Additive White Gaussian Noise (AWGN) channel. The BER results obtained using Matlab/Octave simulation scripts show good agreement with the derived theoretical results.

With Binary Phase Shift Keying (BPSK), the binary digits 1 and 0 maybe represented by the analog levels and respectively. The system model is as shown in the Figure below.

Figure: Simplified block diagram with BPSK transmitter-receiver

Channel Model

The transmitted waveform gets corrupted by noise , typically referred to as Additive White Gaussian Noise (AWGN).

Additive : As the noise gets ‘added’ (and not multiplied) to the received signal

White : The spectrum of the noise if flat for all frequencies.

Gaussian : The values of the noise follows the Gaussian probability distribution function, with and .

Computing the probability of error

Using the derivation provided in Section 5.2.1 of [COMM-PROAKIS] as reference:

The received signal,

when bit 1 is transmitted and

when bit 0 is transmitted.

The conditional probability distribution function (PDF) of for the two cases are:


Figure: Conditional probability density function with BPSK modulation

Assuming that and are equally probable i.e. , the threshold 0 forms the optimal decision boundary.

  • if the received signal is is greater than 0, then the receiver assumes was transmitted.
  • if the received signal is is less than or equal to 0, then the receiver assumes was transmitted.




Probability of error given was transmitted

With this threshold, the probability of error given is transmitted is (the area in blue region):



is  the complementary error function.

Probability of error given was transmitted

Similarly the probability of error given is transmitted is (the area in green region):


Total probability of bit error


Given that we assumed that and are equally probable i.e. , the bit error probability is,


Simulation model

Matlab/Octave source code for computing the bit error rate with BPSK modulation from theory and simulation. The code performs the following:

(a) Generation of random BPSK modulated symbols +1’s and -1’s

(b) Passing them through Additive White Gaussian Noise channel

(c) Demodulation of the received symbol based on the location in the constellation

(d) Counting the number of errors

(e) Repeating the same for multiple Eb/No value.

Click here to download Matlab/Octave script for simulating BER for BPSK modulation in AWGN chnanel.

Figure: Bit error rate (BER) curve for BPSK modulation – theory, simulation


[DIGITAL COMMUNICATION: PROAKIS] Digital Communications by John Proakis

473 thoughts on “Bit Error Rate (BER) for BPSK modulation

  1. Hi Krishna,
    Thank you for this website

    i have two question please
    1- Why the WGN (n) is outside the statement (for) while in other your program (script_ber_bpsk_rayleigh_channel) is inside (for).

    2- Please can you give me the equation of WGN or give me the name of reference

    1. @supas: Replies
      1/ ideally, one would want to give different noise realization for each snr point. with that in mind, having the wgn statement inside the for loop is better
      2/ wgn is created using randn function

  2. Hi Krishna

    I have a question regrading “sigma” the standard deviation of noise:
    We know that sigma = sqrt(No/2), for complex noise. right?
    But how about real noise? Is sigma = sqrt(No)?
    where No is noise power spectral density.
    I look forward to hear from you soon.
    Many Thanks

  3. pls can you help solve this question; Given a stationary transmitter is sending at a center frequency of 900 MHz, the receiver is
    travelling at speeds of (a) 1 km/hr; (b) 10 km/hr; (c) 100 km/hr; or (d) 500 km/hr. Draw
    a graph of four curves that show deviation of spectral frequencies from the center frequency
    for the above four scenarios. Use a legend properly to distinguish different scenarios.
    5. Suppose that a mobile node is moving at a speed of 150 km/hr in a typical urban environment
    (e.g., highway). It transmits a signal to the base station at a center frequency of fc = 1 GHz
    using a binary modulation at a data rate R. It is known that the typical RMS delay spread
     of multipath propagation in this scenario is around 5 s. For each of the following cases,
    decide if the received signal is best described as undergoing fast/slow fading, frequency
    selective fading or flat fading.
    (a) The mobile node is sending at a data rate of R = 1 Mbps.
    (b) The mobile node is sending at a data rate of R = 500 kbps.
    (c) The mobile node is sending at a data rate of R = 50 kbps.

  4. First of all, congratulations for this helpful and nice explanation. I’m dealing with the similar task (maybe the same) these days. I determine two waveforms, Tb and BW, then I’m trying to draw the plot of Pe versus BW. Is yours exactly the same thing that I want to do? However, in my design, there are two integrators. I couldn’t get the point that whether you used integrator in your matlab code or not. Also, if we use lowpass filter, instead of AWGN is there a change? I’m very new at digital communication, so I’m sorry for these bad questions 🙂

  5. Hi ,I need program for BER reduction using SSPA(solid state power amplifer ) model .or else BER reduction using PTS scheme.

  6. Hi!

    You’ve calculated the BER with a message of 1exp6 bits, and the curve obtained is perfectly over the reference, but if you take less than 1exp6 (e.g. 1exp4) apears irregulats peaks on the curve.

    My question is……that’s something that I have to worry? Or it’s correct the response of the curve?

    Thank you very much!


    1. @Yamsha: To get confidence in the simulated results, one need to get sufficient number of errors. For eg, to get a bit error rate of the order of 10^-6, one needs to send atleast 10^7 bits.

  7. Dear Mr Krishna.

    Im trying to draw the same function, BPSK with AWGN, in excel.
    I get a completely different graph, so i must be doing something wrong.

    May i contact you by mail to send you my table?.

    Thanks very much.

  8. Can you please provide the code for 2×2 MIMO with spatial diversity and different reciever combining schemes for a system using m-PPM modulation. Using gamma-gamma channel model.

  9. I need this in matlab with expalnation.
    Compute the error probability of a BPSK as a function of Es/NodB (in decibel)
    {1; 2; 3; 4; 5; 6; 7; 8; 9; 10}.
    1.What is the energy of a BPSK symbol (Es)? And the Energy per bit (Eb)?
    2.The noise variance is
    sigma^2 =N0/2
    hence for each value of Es/NodB you can derive the associated sigma^2 that you
    need to generate your additive white gaussian noise.
    3.Generate a vector x representing N=1000000 BPSK equiprobable symbols.
    4.Estimate the transmitted symbol x based on the received signal y = x + n.
    Show the histogram of y for Es/NodB = 1 and Es/NodB = 10 (comment the
    fi gures)
    5.Compute the average error probability as the number of errors over the total
    number of transmitted symbols N.
    6.Repeat the steps for each value of Es/NodB and plot the corresponding P(e) in
    a log scale (use the function semilogy with “blue” line). Comment the results.
    What is the minimum Es/NodB to guarantee a P(e) < 10^-5? In the same
    gure plot ("red" line) the theoretical P(e) of a BPSK (see slides). Comment

  10. could you please explain the relationship between the BER and the Eb/No and why we get this graph shape
    looking for your reply

  11. hey

    How do you calculate the SNR if given bit error rate of 10^-3.

    Secondly if now i am given a symbol like P which when converted to binary is equal 011. How do i calculate the snr if i need a symbol error rate of less than 10 ^-4 ??


    1. @pawan: you can either use the visual information from the ber plot to find the minimum snr required to hit a given bit error rate. for eg, bpsk in awgn requires around 7dB of Eb/N0 to hit 10^-3 ber. One can also use the erfinv() function.

  12. Hi, Krishna

    Would like to ask how to add attenuation to the transmitted signal?
    which line of code should i edit? is it from ”y = s + 10^(-Eb_N0_dB(ii)/20)*n;”
    In case if attenuation = 0.3

  13. Hi Krishna,
    I want to demo about DSSS spread spectrum systems by MATLAB. it’s my project
    it uses modBPSK and PN
    but i’m not good at matlab programming
    can you help me about it ?
    thankyou very very much !!
    my email :

  14. Hi Krishna,

    In this post, it is very useful that you show the relationship between BER and SNR. Actually, I want to describe the relationship between BER and transmission distance for BPSK in AWGN channel. Please give some advices for this.

    Thank you so much!!

    1. @candy: To convert to a distance, one needs to know – Transmit power, Path loss, Receive noise power
      The SNR, dB at the receiver is = Tx Power, dBm – Path Loss, dB – Rx Noise Power, dBm

      The transmit power is chosen by the designer. The path loss depends on the square of the carrier frequency and square of the distance in free space. Check out
      The receiver noise power depends on the bandwidth of the receiver. For eg, a receiver with a 20MHz bandwidth will have a thermal noise power of -174dBm/Hz + 10*log10(20e6) = -101dBm.

      Hope this helps.

      1. Dear Krishnan,
        Thank you so much for your help!
        I have another problem that I can not understand as:
        I used the formula: BER = 0.5*erfc(SNR) for BPSK in AWGN channel to plot the picture which show relationship between BER and propagation distance through the relationship between SNR and distance based on free space path loss model. But I can not explain why BER can not be greater than 0.5 even the distance is increased up top very large value. I think that this result is not suitable with the fact that when BER will be increased to 1 when distance increased from 0 to infinite.

        I hope you give me some advices for this.

  15. Dear Krishnan,
    Could you help me writing the code for this question.

    Simulate 4-PAM (k = 2,M = 4). Pairs of bits are mapped into symbols s, where s belongs to the alphabet S = (3A, A,-A,-3A). Note that the input symbols are equiprobable, and there is no need to generate individual bits. Additive noise n is real AWGN process (same as in (1)). The input into the detector is r =s + n. The
    detector finds the closest point in S to r. The average symbol SNR s = 5A square / 2. The bit SNR b = SNR s / 2. Simulate performance of this system for SNR b = 7, 8, 9, 10, 11 dB and find the symbol and bit error rates.

    Your cooperation in this regard will highly be appreciated


  16. I want the code for IEEE project of “Error correction between convolutional codes using Chaotic switching.

  17. Hello Sir
    I am new here and i need your help.Please give me answers of my few questions. I will be thankful to you.:
    1. ip = rand(1,N)>0.5
    how it generate only +1 and -1
    what is the concept of using the term>0.5

    2. Eb_N0_dB = [-3:10]
    how the range of Eb/No is chosen and what is the need of this?

    3. for ii = 1:length(Eb_N0_dB)
    what is the concept of this function?

    1. @khushi: my replies
      1. rand() generates a uniformly distributed number in the range from [0 to 1). The >0.5 allows to generate a input sequence with approximately equal probabilities for 0 and 1.
      Then 2*0 – 1 = -1 and 2*1 – 1 = +1

      2. This is the region where the BER for BPSK modulation changes from from very high ber (>0.1) to very low ber (<10^-4)

      3. To find the ber for different Eb/N0 values

      1. Thank you very much Sir for solving my problems
        One more thing I want to know that how do we know that in which range BER changes from from very high ber to very low ber.

  18. Hi Krishna,
    Could you answer the following q’s please if you have the time? Thanks. Zoe

    1. I don’t know how the following counts. Is nErr(ii) an array?
    And I don’t precisely know what ‘size’ and ‘find’ functions do?
    (I know ip-ipHat = 0 is no error)

    nErr(ii) = size(find([ip- ipHat]),2);

    2. In the following is the decimal version of EbNo getting multiplied with n, mean of the Gaussian noise? (of course then added with s, -1 or 1, depending on what was sent)
    y = s + 10^(-Eb_N0_dB(ii)/20)*n

    3. The following is confusing. Why two real and imaginary component Gausian functions added and normalized by inverse of “sqrt(2)”
    Could you write the formula or equation it is derived from?

    n=1/sqrt(2)*[randn(1,N) + j*randn(1,N)]

    1. @Zoe:
      1. nErr is the count of errors.
      find() finds the index of elements which are different between ip and ipHat
      size() counts the number of elements which are reported by find()

      2. it is the Eb/N0 in linear scale applied to the noise voltage to produce the desired Eb/N0 ration

      3. one can possibly send information on the two dimensions together. Hence we model as two randn() variables each with variance 1/2. Hence the normalization by 1/sqrt(2) on the voltage signal.

  19. Hi dear,

    how are you.I need a help from you because I have a project and there some points i don`t understand it.

    It is about the square root raised cosine pulse ,we need to plot it with different roll-off factor(B=0.25,B=0.5,B=1.
    After that consider a binary antipodal signaling over an ideal AWGN channel at the normalized bit rate of 1 bit second where TB =1
    x(t)=+or-p(t) +w(t)
    where w(t) is an additive white gausian noise ,simulate model for this system and plot the resulting BER (B=0.5)

  20. Hello Krishna Pillai,
    thanks alot for the wonderful work you did here.please can i have d privilege to have your personal email address. I have something personal to email to you with respect to my undergraduate project work.Thanks.

  21. Please help me with simulating BPSK with direct-sequence spreading using m-sequence .I have tried doing the spreading and the despreading as well as noise addition but the BER vs. EbN0 curve is higher!!!! than the theoretical one ,which is the same as BPSK without spreading .
    the problem could be with the addition of the noise ; I did multiply the transmitted chips by 1/sqrt(N) ,where N is spreading factor ,in order to make sure that the bit power is 1 .

    Here is my code:
    b = 10000; % number of bits or symbols

    %—————– Transmitter————–
    %Message signal
    ip = rand(1,b)>0.5; % generating 0,1 with equal probability

    %generating m-sequence
    polynomial=[6 1 0];

    signal = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 1 (polar)
    mseq=2*pnSequence-1; % PN sequence in polar form

    %spreading message signal ‘signal’ using pn sequence ‘mseq’,the processing gain
    %’N’;length of the PN sequence.


    n = randn(1,N*b) ; % white gaussian noise, 0dB variance
    Eb_N0_dB = [-3:10]; % multiple Eb/N0 values

    for ii = 1:length(Eb_N0_dB)
    % Noise addition
    y = spreadedSig + 10^(-Eb_N0_dB(ii)/20)*n; % additive white gaussian noise

    despreadedSig= despreadedSig’;

    % receiver – hard decision decoding
    ipHat = real(despreadedSig)>0;

    % counting the errors
    nErr(ii) = size(find([ip- ipHat]),2);


    simBer = nErr/N; % simulated ber
    theoryBer = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))); % theoretical ber

    % plot
    close all
    hold on
    %axis([-3 10 10^-5 0.5])
    grid on
    legend(‘theory’, ‘simulation’);
    xlabel(‘Eb/No, dB’);
    ylabel(‘Bit Error Rate’);
    title(‘Bit error probability curve for BPSK modulation’);

    end of code

    So please can you check what is wrong with my code??

      1. @Krishna: Hello Krishna
        I have tried the no noise case and gave me zero BER . I couldn’t figure out what is the problem with the scaling ??!! Do you have any code for spreading and despreading ?
        I need the correct coe for my final project on my Bachelor degree. So I should be very grateful if you can help me with this.

  22. pleas Mr I would like to help me to find calculation BER vs SNR when using 64DAPSK modulation with DWT-OFDM in simulation MATLAB, thanks.

  23. Your article really open my eyes.
    Is there anyone about calculate theory BER value of frequency selective channel in your BLOG?
    Yushi Chen

  24. i want code on CDMA transimeter and reciver with AWGN for multi user by LLR log like ratio in receiver iterative decoding ,…….i make it but without LLR can u help me plz also i put spreader but not coded as block thx

  25. Dear Mr. Krishna
    Would you help me in my project!!!
    I study this paper in IEEE for satrting my thesis in master degree. “Cross-Layer combining of adaptive Modulation and coding with truncated ARQ over wireless links” .1343909 abstract
    This paper appears in: Wireless Communications, IEEE Transactions on
    Issue Date: Sept. 2004
    Volume: 3 Issue:5
    On page(s): 1746 – 1755
    but I have som misunderstand whith this paper and manye other papers like this .
    The first problem is finding average SNR in mode(n).
    I don’t know how can I select the average SNR in (7), (8),(15) ,…
    and the second problem is ,if we select the SNR whit gama distribution in matlab for example:
    then gama average is: ab (in gama distribution).
    In fig 4, 5 ,… we have average SNR in horizontal axis and I don’t know how select from 0 to 30 db
    best regards

  26. Hi Krishna,
    your coding’s were very useful for my simulation lab works..could you please send me the comparison of various error correcting codes for a particular modulation scheme..

  27. Thank u very much Mr.Krishna.. Could you please name a few companies recruiting in this field… I knew only the IT companies no idea about this type of companies and what will the requirements to get into those companies……

  28. Hello Mr.Krishna

    How to get rotated Pi/4 QPSK constellation? I got QPSK constellation how to get Pi/4 rotated constellation for QPSK

  29. I am having problems simulating the BER vs SNR curve for the binary on-off keying modulation. My plot looks backwards. Do you have any suggests.
    Here is my code
    % EE474 Lab 1

    clear all;
    close all;

    Es =1;
    count0 = 0;
    %Simulated BER
    for snr =0:7
    count0 = count0+1;
    error =0;
    right = 0;

    snr_number = 10^(snr/10);
    N0 = Es/snr_number;
    count = 0;
    else d=0;
    S = d*sqrt(Es);
    noise = randn*sqrt(N0/2);
    D =S+noise;
    if D>0
    d_hat = 1;
    d_hat =0;
    if d == d_hat
    right = right+1;
    error = error+1;

    ber_sim(count0) = error/count

    hold on;
    ylabel(‘BER (dB)’);
    title(‘BER vs SNR BPSK’)

    Your help is appreciated.

  30. Dear Mr. Krishna

    I’d be grateful to you if it’s possible to provide single carrier and OFDM code (Matlab) for both timing and frequency offsets with and without cyclic prefix (both for single carrier and OFDM).

  31. Gud morning Mr.Krishna thank u for posting ur valuable comments..

    for i = 1:length(SNR_dB)
    y = awgn(s,i);
    y_hat = real(y)>0;

    I checked this one no need for ‘ber2’.. I used ‘ber’ as an array to store the values of bit error ratio since biterr is an inbuilt fuction and [n,b] where ‘b’ bit error ratio for a particular snr value and we need Bit Error Ratios for all differen SNR values, for that I used variable ‘ber’ to store the values of ‘b’

      1. Yes Mr.Krishna…
        I’m currently pursuing my M.Tech(Communication Engineering) in VIT.. I’m willing to work in wireless field.. I need ur guidelines to work in communication field or in signal processing so that I can prepare for my future… Pls do reply waiting for ur precious reply.. Thank You

        1. @Thiyagi: All the best for your studies. With out being specific, I believe having the following knowledge will help you in good stead when trying to chart out a career in wireless communication/signal processing :
          a) Understanding of RF architecture, RF impairments and it’s effect on the system performance
          b) Phase/frequency syncrhronization
          c) Channel effects – channel estimation/tracking
          d) Channel coding
          e) Multi user communication … to name a few…

      2. hi,i need to have m file of each modulation and the figures of error in matalb without using the order of modulation that are in matlab.
        can you help me?

  32. Hello Mr.Krishna Pillai I tried to plot the BER for BPSK using the following code but didn’t get correctly..Could you please kindly correct me what went wrong..
    clear all
    close all
    x= randint(1,200);
    y= [2];
    k= log2(y);
    z= reshape(x,length(x)/k,k);
    s=pskmod(d,M);%Psk modulation

    for i=snr;
    q=pskdemod(p,M);%psk demodulation


      1. Yes mr.Krishna i’m getting bit error rate as zero without adding noise i.e not used ‘awgn’ function to add noise

        1. Mr.Krishna for the below code i’m achieving same BER at less SNR comparing to the theoretical value, since i used AWGN inbuilt function to add noise..Whether this approach is correct? With the program given in this website it works good, from this program I tried with my logic just want to know whether the below logic is correct..Thank u
          close all;
          clear all;
          N = 10^6; % number of bits or symbols
          x = randint(1,N);

          for i=1:length(x)
          y1(i)=1; y11(i)=-1;

          if x(i)==1
          if x(i)==0
          n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)];%AWGN
          SNR_dB = [-3:35];
          for i = 1:length(SNR_dB)
          y = awgn(s,i);
          y_hat = real(y)>0;
          form_ber = 0.5*erfc(sqrt(snr)); % theoretical ber
          hold on
          legend(‘theoretical BER’,’practical BER’)
          axis([-3 10 10^-5 1])
          grid on

          1. @Thiyagi: The code looks ok. You can check couple of points:
            a) awgn : whether the noise power is as expected. Maybe you can try with ‘measured’ option.
            From Mathworks
            “y = awgn(x,snr,’measured’) is the same as y = awgn(x,snr), except that awgn measures the power of x before adding noise.”

            b) Counting the number of error. That piece of code looks funny. Did not understand the need for ber and ber2.
            Should be easy for you to figure out.

  33. hai,
    i would like to know why in y = s + 10^(-Eb_N0_dB(ii)/20) we use a negative sign her e for eb no.
    Thanks in advance

  34. 1.10^(-Eb_N0_dB(ii)/20)*n

    i want to answer the confusion of /20 why not by /10?

    In fact sir Krishna Pillai want to make noise variance equal to 1 or 0dB.
    you can check by command max((10^(-Eb_N0_dB(ii)/20)*n))= 1.01 where as
    max((10^(-Eb_N0_dB(ii)/10)*n))= 0.3251 so its clear scaling…..remember before these commands i just removed the complex part i.e n = 1/sqrt(2)*[randn(1,N)]; for my easyness.

    I also noticed that this scaling change for system to system e.g for non coherent 4FSK -it becomes like (10^(-sqrt(2)*Es_2N0_dB(ii)/20))*n becoz in 4FSK i need variance i scaled it to 0.5 by just multiplying by sqrt(2)

  35. Do you have code on how to generate a NRZ signal?

    And do you have code on how to generate the eye diagram. Thanks.

  36. Hi krishna,

    Could i have codings with different modulations with different subcarriers for any one user in MIMO OFDM system .output shouild be throughput versus time or ber and time.please help me sir.Thank you in advance

  37. Hello Krishna Sankar,

    I am doing some research on chaotic carriers and I was planing to make this BER for comparison.
    Your post save me a lot of time to focus on the real subject of my work.
    Thank you very much for the help!

    Regards from Brazil
    Marcos Amaral

  38. Hi Krishna,

    I am working on different modulation technique. I am badly needed BER code of QPSK, (QAM 16,64)with AWGN channel.I have to submit my paper on 30may2011.

    please send me code , this very helpful for me.


  39. Dear krishna
    impressive works! hay, can you provide me simulation codes for BPSK, QPSK, 16QAM & 64QAM with AWGN channel in OFDMA modulation? these would be very much helpful for my thesis. thank you.

  40. Hello Krishna,
    I have a question.. I am hoping to design a OFDM communication system…
    If you were to design an OFDM communication system.. what and where would u start from?

  41. Hi Krishna,

    what about 8-PSK ?
    is it the same to 16-PSK and just change M=8 only?
    or there is other things that i should to change ?
    becouse when i changed M to 8 i got an output and no errors
    pls answer me it’s very importanat
    thanks alot man

  42. hi
    i m doing graduation we have a project in which we have to design the communication sysem in which data+modulation(bpsk and 8-psk)+awgn then threshold detector and then we had to compute bit error rate.. can you help me in this…..

  43. dear krishna,
    i need to simulate BPSK modulation to analyse the bit error rate in AWGN channel and reyleigh fading channel for different SNR.the carrier frequency should be 20MHz.if u hav matlab codes for dis,help me.m running out of time.
    xpecting fruitful reply.

  44. Thanks for your detailed post!
    But I have question about the comment of this line in your codes
    s = 2*m-1; % BPSK modulation 0 -> -1; 1 -> 0
    I think maybe you want to change original 0 to -1, but 1 should be no change, is it?

  45. Oops i am sorry if this is a total blunder.. In digital modulation techniques we modulate a sine or a cos wave using bits and transmit them as sine or cos waves.. But in this code for BPSK modulation I never found the usage of such things.. could you please clarify my doubts in simple words… Regards Ajith

    1. @Ajith: No, its not a blunder at all, tt’s a genuine query. There are two aspects to the reply.
      a) In the case of baseband transmissions, we send the information on pulses and in the most simplest case, we send out rectangular pulses of varying amplitude to convey the information (or we can do some better pulse shaping to reduce the out of band emissions, while not introducing ISI)
      b) In the case of pass band transmissions, we do the pulse shaping, then upconvert to a carrier and then transmit.

      In both (a) and (b), for computing BER, the results which we get even if we do not simulate pulse shaping + up conversion is same as the one without it (assuming, we undo the operations optimally). Hence did not put those in place.

      Further, I have written a post doing BER computation with rectangular pulse shaping with matched filtering

  46. hey krushna
    That really helpful for me.
    but i need it’s paper too. can u send it to me plzzz

  47. hello sir
    can you please guide me,how can i plot BER error rate of uncoded and coded BPSK
    i know the formula of BER for Uncoded BPSK ,i have plotted it,but I am really cunfused about the Coded Bpsk,I am using BCH codes which are block codes
    Waiting for your reply

  48. hello sir …
    in BPSK constallation we map bit 1 to sqrt(Eb) and 0 to -sqrt(Eb). these are real points then why to add complex noise …we can directly add real noise of unit variance as follows…
    N = 10^6 % number of bits or symbols
    rand(‘state’,100); % initializing the rand() function
    randn(‘state’,200); % initializing the randn() function

    % Transmitter
    ip = rand(1,N)>0.5; % generating 0,1 with equal probability
    s = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 0
    n = randn(1,N) ; % white gaussian noise, 0dB variance
    Eb_N0_dB = [-3:10]; % multiple Eb/N0 values

    for ii = 1:length(Eb_N0_dB)
    % Noise addition
    y = s + 10^(-Eb_N0_dB(ii)/20)*n; % additive white gaussian noise

    % receiver – hard decision decoding
    ipHat = real(y)>0;

    % counting the errors
    nErr(ii) = size(find([ip- ipHat]),2);


    or we can use awgn function to pass through awgn channel which will add real noise if data is real and will add complex noise if data is complex… if i proceed in that way the performance is not matching with the theoritical analasys… can u explain me clearly why should we add complex noise…
    expecting early reply….
    Thanking u sir

    1. @rama krishna: Adding complex noise does not affect the end result (as we are ignoring the imaginary part at the receiver). However, just make sure that scaling is correct. You should multiply factor of 1/sqrt(2) to the noise term which you have defined.

  49. Hi sir,
    I want some programmes on DAPSK(differential amplitude phase shift keying) in OFDM systems.Is there any matlab codes in DAPSK.Please reply me as soon as possible.

  50. Hi Krishna Pillai ..
    I whant to simulate BER for BPSK but for 5 or 6 user not for 1 user
    what is the changement applicated in this programme.

  51. dear sir Krishna Sankar
    I am very happy to find your blog … and found what I was looking for … I am a student majoring in telecommunications engineering and was doing the final tasks of rain attenuation, said supervisors told to discover how much value other than tresshold BER for MQAM rmodulasi … I want to ask how can I download the book John Proakis as an accurate reference book saya.Mohon bantuanya final … and also how the graph of the BER curve modulation 16…………

  52. pls i need a matlab program for signal to noise ratio in a wireless communications system.. how do i do the write up for the matlab program??

  53. your kind guidance on MATLAB has been really a great help for MATLAB starters. hope u will continue in future too.

  54. Hi,
    Can anyone help me in my project!!!
    i Have read here that there is no difference between BER vs SNR in case of BPSK and OFDM using BPSK.
    But my project guide has told me to mathematically prove that BER vs SNR is better for OFDM using BPSK than simple BPSK.
    Suppose there is N no. of subcarrier.Then for each subcarrier there will be BER.But if i want to calculate the total BER for n no. of sub carrier then how the BER of indivisual subcarriers are related???

  55. hello sir,
    m doing ma project on ber calculation using matlab simulation for ofdm transmission for multipath wireless communication so please if u cud help me out in knowing which book wud b d best 2 go through….where i wud b getting much of d information regarding d basics of ofdm , transmission n reception of signls using ofdm, n n… ber calculation n its various graphs hw 2 plot dm n al….
    please do help me out sir in dis ……m very confused abt dis…..

  56. hello sir,
    please give us an idea how to analyse matlab code for ofdm-awgn channel with that of theory. what are the pilots and why are they used?

  57. Hi There,

    I need to design and develop a simulation platform in C/C++ to simulate the performance for BPSK- and 8PSK modulation through an AWGN channel and evaluate the biterror rate (BER) performance for BPSK and 8PSK in the range Eb/N0 2 [−4, 8] dB.Can you help me with any suggestion or source code.

  58. respected sir
    please can u help me in providing matlab code for rs codes using interleaver &qam in awgn channel
    i will be thankful to u

  59. hi,

    I would like to demodulate QPSK but using soft decision. can you help me or guide me how I want to do this? I try the matlab function using

    demodh= modem.pskdemod(ht, ‘outputType’, ‘bit’,’DecisionType’, ‘LLR’, ‘NoiseVariance’, sigma);

    but the bit that I’m receive sort like it have inverse sign.
    for example: if I transmit bit : 1; 0; 1 ,
    I received : -10.8; +9.7; -11.2.
    I’m expecting that : +10.8;-9.7;+11.2.

    Or is it LLR gives this inverse sign? TQ so much if you can help me.

  60. Hi krishna!

    I need matlab code to obtain BER vs SNR curve for binary on-off keying.
    also ,code for generating SER vs SNR curve for 3ASK modulation

  61. Dear Krishna,

    I already ask this question please. reply
    Can you little bit tell me how to simulate the Uncoded BER and ergodic capacity for webb channel using QPPM modulation.

  62. Dear Krishna,

    Can you little bit tell me how to simulate the Uncoded BER and ergodic capacity for webb channel using QPPM modulation.

  63. Hi Krishna,
    I know the BER of BPSK is 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))),
    if I put channel coding before the modulation,and the code rate is R,
    then if the BER equals 0.5*erfc(sqrt(R*10.^(Eb_N0_dB/10)))?
    then BER with code rate 1/2 is higher than 3/4,but acturally ,it should be
    lower. where is my mistake ?

    Hoping for your help, thank you


  64. please can u help me in this project ::::–

    2) final report(OFDM):

    * genearte large no of bits using randint

    * modulate the data once with bpsk & then 16-QAM

    * bpsk mod is 2*data-1

    * after modulation u should loop on the data with step of 64’s

    each time calculating their ifft

    * add the noise to each 64

    * if u remeber how to add noise then do it as u know,
    else use awgn

    sigandnoise=awgn(signal, SNR in dB , ‘measured’ )

    * do fft with steps of 64’s , the same groups of symbols that were ifft’ed

    * demodulate the data ( if bpsk then bits= recdata > 0)

    *calculate ber using
    ber=sum(xor( transmitted bits , received bits ) )

    * the whole program should be done in a large loop that loops over

    SNR -5 -> 15

    * in case of fading
    multiply with channel before ifft then divide after ifft

    * multiplication is done using (.*) not just (*)

  65. dear krishna…

    From my past question, actually i got my result and understand..but..i have a problem with my coding…can you help me please…
    My coding…
    %directional antenna

    %power noise
    k=1.38e-23; %k=Boltzman’s constant
    t=300; %t=noise temperature in Kelvin
    b=5e6; %b=bandwidth

    %directional antenna



    N = 10^6; % number of bits or symbols
    rand(‘state’,100); % initializing the rand() function
    randn(‘state’,200); % initializing the randn() function

    % Transmitter
    ip = rand(1,N)>0.5; % generating 0,1 with equal probability
    s = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 0
    n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % white gaussian noise, 0dB variance
    % Eb_N0_dB = [-3:10]; % multiple Eb/N0 values
    % k=1:1:10;
    % for ii = 1:length(k)
    % % Noise addition
    % yomni = s + 10.^(-k(ii)./20)*n; % additive white gaussian noise
    % % receiver – hard decision decoding
    % ipHatomni = real(yomni)>0;
    % % counting the errors
    % nErromni(ii) = size(find([ip- ipHatomni]),2);
    % end

    for ii = 1:length(ebno101)
    % Noise addition
    y10 = s + 10.^(-ebno101(ii)./20)*n; % additive white gaussian noise

    % receiver – hard decision decoding
    ipHat10 = real(y10)>0;

    % counting the errors
    nErr10(ii) = size(find([ip- ipHat10]),2);


    simBer10 = nErr10./N;

    grid on
    xlabel(‘Eb/No, dB’);
    ylabel(‘Bit Error Rate’);
    title(‘Bit error probability curve for BPSK modulation in simulation’);

    when i changed the Pt=10:10:100; actually i got the result..can you help me pliz….

    thanks a lot…

      1. dear krishna…

        In theory, when i applied my SNR into your coding, i got the result.
        But, when i applied my SNR into your coding for simulation, i got the problem. I think that problem occurred at SNR or BER. But i dont know where my coding is problem? Can you help me pliz…
        Thanks a lot..

  66. Hello friends: i am working on MAP , ML decoding for convolutional there any one who too work on similar field ..?

  67. Hi Krishna,

    Actually, im studying about your coding for BER vs SNR for BPSK modulation. For simulation, when i changed the value for EbNodB = [-3:10] to [1:10], i cant get the result. why this problem happened?

  68. hi krisna,
    i hope you are fine.plz could you send me BPSK,QPSK,16QAM,64QAM modulation and demodulation simulation in matlab and simulation of adaptive modulation of convolutionaly coded for BPSK,QPSK,16QAM,64QAM?please help me.

  69. hi sir
    i have problem in BPSK with RS codes…..when i/p is given to pskmod function it says that “does not support complex airthmatic”.however i/p given is in 1 0 form

  70. Please guys I am having trouble writing a code for non coherent detection, I am required as part of my project to Assume Rayleigh fading channel with BPSK modulation. Using MAT-LAB plot bit error probability (BEP) under non-coherent de-tection. Your ¯gures should include plots from both analysis and simulation.Use average SNR (complex) from -5 to 20 dB.

    Thanks a lot, i will be very grate ful to get help from you guys.

  71. hi

    Why you are using 10^(-Eb_No_db(ii)/20) the minus sign in this formula because for voltage scaling it should be 10^(Eb_No_db(ii)/20).

    1. @Hamid: The negative sign came as I am scaling the noise voltage. I am keeping the signal swing the same and reducing the swing of noise voltage to simulate various Eb/N0 values.

  72. I am working on an adaptive modulation model on Simulink. For that first I am trying to get probability of error rate vs snr of different modulation techniques. On modelling 16 QAM modem I have problem plotting its graph by getting bit err rate from simulink model and performing graph plotting on matlab. following is the code im using in matlab..

    clear; clf;
    M=16; % for simulink

    for i=1:length(snr)
    grid on

    please guide what is the error in this code… thanks

  73. Hi…
    I need to Simulink Probability of error for PSK, 16QAM & MFSK. but don’t know how to simulink the equation: Pe=0.5*erfc(sqrt(snr)).
    I tried to call the fuction from Matlab using Embedded Function Block in Simulink. But, the program din’t work. Can u suggest me a Block for finding Pe or a Program to call from .m file..?

  74. Hi Krishna,

    I was working on a IEEE paper titled”Xiaodi Zhang and N.C. Beaulieu, “A Closed-Form BER Expression for BPSK Using MRC in Correlated CCI and Rayleigh Fading,” IEEE Trans. Communications, vol. 55, no. 12, pp. 2249-2252, Dec. 2007.”.
    Since you have also worked on similar field, I hope u can help me..

    I was stuck with re creating the 1st fig in the paper.
    SO, can you please look into it and help me out..


  75. Hi krishna
    Pls in essence how and where do we set the different values for EbNo/ or S/N or SNR to achieve different BER plots using your simulation example. Pls i will appreciate ur reply.


  76. Hi krishna
    Pls in essence how and where do we set the different values for EbNo/ or S/N or SNR to achieve different BER plots using your simulation example. Pls i will appreciate ur reply.


  77. Hi Krishna.
    Thank you very much. God will bless you. Below is the answer you gave me for my concern on this formular 10^(-Eb_N0_dB(ii)/20)*n
    “Do not change the division factor. Rather change the value of Eb_N0_dB. The division by 20 is required to convert dB into voltage”.
    Please give me more direction, by giving me an example of what you mean by changing the VALUE of Eb_No_dB. or how do we get the value
    Am thinking is Eb_No_dB =[0:10], [0:20], [0:30] and so on, but am not very sure. Pls, you are indeed a very good teacher, help me out

    Thanks, you are really very helpful, you must really reap the fruit of your efforts.

  78. Hi Krishna.
    Thank you very much. God will bless you. Below is the answer you gave me for my concern on this formular 10^(-Eb_N0_dB(ii)/20)*n
    “Do not change the division factor. Rather change the value of Eb_N0_dB. The division by 20 is required to convert dB into voltage”.
    Please give me more direction, by giving me an example of what you mean by changing the VALUE of Eb_No_dB.
    Am thinking is Eb_No_dB =[0:10], [0:20], [0:30] and so on, but am not very sure. Pls, you are indeed a very good teacher, help me out

    Thanks, you are really very helpful, you must really reap the fruit of your efforts.

  79. Hi krishna, I have actually read the answers given to you by the concerns generated by the formula below
    for further clarification, i have actually divided by 18,16,14,12——3 to get different plots that actually decreases down the graph . Want to know if is how one can plot for BER with variable noise in the channel. I will appreciate if you reply me soon. Thanks

  80. Hi krishna, I have actually read the answers given to you by the concerns generated by the formula below
    for further clarification, i have actually divided by 18,16,14,12——3 to get different plots that actually decreases down the graph . Want to know if is how one can plot for BER with variable noise in the channel. I will appreciate if you reply me soon. Thanks

  81. Hi Krishna,

    thanx a lot for your website

    i have a simple question:

    what about the unequally probable, e.g. when P(s0)=0.25 & P(s1)=0.75 ??

  82. Thank you for this website ,it is awesome
    could you help me to know what is the role of the following

    rand(‘state’,100); % initializing the rand() function
    randn(‘state’,200); % initializing the randn() function
    thank you

    1. @christine: The random numbers generated by the program can be initialized to enable us to run multiple simulations with the ‘exact same’ random numbers ;). Setting the state of the rand() or randn() enables one to do so.
      Use >>help rand or >> help randn to get more information.

  83. This Krishna Pillai guy is a great help and his website undoubtedly very good resource on communication system topic. I got necessary informations about Vitervi Algorithm, Hard and soft decision.

    Ajay Pratap Singh apts

  84. Dear Krishna:
    I knew that OFDM technique, improve the BER performance in frequency selective fading channel, Can you help me in writing code using matlab program show that , I have probem in how write code for frequence selective fading.
    Thank you .

  85. I knew that OFDM technique, improve the BER performance in frequency selective fading channel, Can you help me in writing code using matlab program show that , I have probem in how write code for frequence selective fading.
    Thank you .

  86. hello Krishna Pillai!
    I’m a student in Ho Chi Minh City, Viet Nam. I’m searching about “the impact of real channel over MiMo system”. your code is useful. So, we assume that receiver is unknown h matrix and we must explore channel by channel estimation. Do you help me?
    Hope you will response me!

  87. Hi Krishna,

    I am working on Forward error correction codes. I have plotted the BER performance curve for Reed solomon codes using BPSK modulation over an AWGN channel from EbNo 0 – 1 . However, the curves are overlapping at the begining between EbNo values 0 and 1 and also in the middle.

    Also the curves for convolutional codes overlap at the begining between 0 and 1. and also in the concatenated codes.

    Could you please give me a detailed explanationc on the reason why the overlapping occurs and the significance.

    The RS codes used were RS(255,191), RS(255,223), RS(255,239) and RS(255,247).
    Thanks, hope to hear from you soon.

    1. @Tito: Firstly, a clarification. When you mentioned about “curve-overlapping”, I guess you meant that – “with coding, the BER becomes poorer than no coding for very low values of Eb/N0”. Agree?

      At-least with respect to convolutional codes, I think it can be attributed to the large number of errors in received coded bits, which makes the Viterbi algorithm go haywire.

      1. Thanks that is exactly what i mean. Why does this occur.
        And i also have an curve RS curve with and without an interleaver, these curves are overlapping at two points. Why does this occur.

  88. Hi Pillai

    I would like to ask you a simple question. I am trying to make a simple simulation using a matlab. It is about that one node using SF(spreading factor)=2, and another node using SF = 8 (orthogonal to each) are encoding the data and sending to one node (asynchronous network) with same power but from different distance at the same time. how can i simulate it? when decoding at the receiver, it uses same ovsf code as the node using SF = 8.

    Thanks in advance.

    1. @Alex: Well, let me try to write the pseudo code for this case
      y1 = x1*code1; % code with SF=2
      y2 = x2*code2; % code with SF=8
      n = noise; % AWGN noise, lets take mean=0, variance = 1
      r1 = y1*scaling1; % scaling factor for first signal
      r2 = y2*scaling2; % scaling factor for second signal
      r = r1+r2+n; % receiver collects both signals corrupted by noise
      x2Hat = r*code2 % receiver tries to correlate r with code2 to recover x2.

      Makes sense?

  89. Hi, my english is not very good looking, but…what is the noise variance for 16QAM and 64QAM?, AWGN channel.

    1. @Diego: The noise variance is independent of the modulation scheme. However, I think you might be looking for error rate for different modulation schemes for a given value of noise variance. You may refer posts:

  90. hi
    i need matlab code for adaptive modulation from BPSK,QPSK,16QAM,64QAM

    so if you have not such code, i hope to guide me how to make it , as you know for low SNR we use like BPSK and for high SNR use like 64QAM

    so how to make this via matlab?


  91. Hi, I am want to plot the pdf for various SNR(Symbol Error) for various modulation schemes. Do you have matlab code for it.

    The derivation in this link is SER Vs SNR, How to change the SER to PDF(of SNR)?

    Thanks in advance

  92. Thank you for your effort…
    i have some question
    1. y = s + 10^(-Eb_N0_dB(ii)/20)*n;
    2. theoryBer = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10)))
    i don`t know why Eb_N0_dB(ii) has minus value…
    At first term,^(-Eb_N0_dB(ii) is divided by 20 but, at second term Eb_N0_dB is divided by 10.
    Why Eb_N0_dB is divided by diferrent value??
    My question is very poor, nevertheless i will wait for your comment
    Have a nice day

    1. @Moon: My replies
      1/ Before this scaling, the signal term s and the noise term n has the same variance of 1. The term -Eb_N0_dB reduces the noise variance
      2/ The scaling is happening on the noise voltage signal, hence the factor of 1/20. In the theoretical equation, note that there is a square root term outside the 1/10 factor. This is equivalent to having 1/20 factor. Agree?

  93. Can someone help me out with a Matlab code for this.
    For ISI chanell with real coeffnts and BPSK modulation write a Viterbi algo decoder with decoding delay >0 (say ‘delta’)

    plot BER vs SNR for varying ‘delta’s

  94. Can anyone help me out with a matlab code for this.
    For ISI channel with L real coeffnts and BPSK modulation write a Viterbi algo decoder with decoding delay >0

  95. i wanna know what does this statement exactly do
    ipHat = real(y)>0;

    is it equivalent to
    if real(y)>0


  96. Hello
    i just want to ask about this statement in the code
    ipHat = real(y)>0;
    what exactly it do
    is it equivalent to
    if real(y)>0


  97. please can you tell me what does this instruction do exactly

    ipHat = real(y)>0;

    is it equivalent to
    if (real(y)>0)
    ipHat = 1
    ipHat = 0


  98. Hi All,

    Can any one guide me to good source…
    How to calculate Bit error probability for given SNR, any modulation technique with error correcting codes(convolution coding).

    Please give some information on this.


  99. Thanks Krishna for reply.

    I want to plot BER vs Eb/No for MPSK and MDPSK. Can u help me in this regard ?

    Also do u have code for FSK coherent and non-coherent with corresponding BER vs Eb/No plot?

    Your early response will b highly appreciated.

  100. first of all, thanks a lot for ur job, it´s really useful for all the DSP internet communty.

    here my question, I´m carrying out some simulations on simulink. modulating BPSK, spreading w/ gold code, going through awgn, despreading and demodulating BPSK, and at the end I measure the BER.

    using simple BPSK mod/demod provided in the communications block set, I´ve got a relative high BER (.5), then I found the so-called “Real BPSK mod/demod” which basically a “real-imag to complex”/”complex to real-imag” block is added to the BPSK mod/demod, and the resulting BER is around .1.

    I´ve tried to find unsuccessfully an explanation to this on technical literature.

    maybe you know something about that.

    Thanks in advance.

      1. thanks Krishna.

        Actually i need matlab code for non-coherent detection of either bpsk or dpsk and also corresponding BER vs Eb/No plots.


        1. sorry. in addition, i want noncoherent detection of bpsk or dpsk for both AWGN AND RAYLEIGH channel.


          1. Thanks Krishna for reply.

            I want to plot BER vs Eb/No for MPSK and MDPSK. Can u help me in this regard ?

            Also do u have code for FSK coherent and non-coherent with corresponding BER vs Eb/No plot?

          2. @RAO: My replies;

            1/ I have written a post on symbol error rate and bit error rate for 16-PSK.
            The equation and code should be easily adapted to M-PSK case.

            2/ I have not tried modeling MDPSK.

            3/ I have written a post on coherent demodulation of FSK

            Hope this helps.

          3. please can you tell the differences in the code if we made it such that the noise is only Gaussian noise with 0 mean and No/2 variance

          4. please can you tell me how that is discussed if you write this statement 0db variance
            in this line of code

            n = 1/sqrt(2)*[randn(1,N)] %+ j*randn(1,N)]; % white gaussian noise, 0dB variance

          5. really , please can you illustrate me which part exactly in the code do this thing as the part the i see in the code is

            n = 1/sqrt(2)*[randn(1,N)]; %+ j*randn(1,N)]; % white gaussian noise, 0dB variance

            and you write as comment that the noise is 0 db variance

            thanks for your great efforts

          6. Hi Mtech project is Cs-OFDMa ,here we are using LAS (smart codes),pls let me know if hav any matlab simulation for the subject mentioned.

          7. Hi Krishna,
            I just read your answers in this website.
            I am working in convolutional codes with 8psk and AWGN. I have some doubts in Iterative decoding. Can you help me.

          8. Hi Krishna,

            I am working on a IEEE paper titled”Xiaodi Zhang and N.C. Beaulieu, “A Closed-Form BER Expression for BPSK Using MRC in Correlated CCI and Rayleigh Fading,” IEEE Trans. Communications, vol. 55, no. 12, pp. 2249-2252, Dec. 2007.”.

            Since you have worked on similar field, I was hoping you can help me.

            I was stuck with re creating the 1st fig from the paper.
            Can you please look into it and help me.


  101. Dear Krishna, thanks for your codes. I am plotting BER for BPSK and QPSK but i am not getting same BER curve. As it should be same for both. I know that the plot should be BER vs EbN0. But how to get EbN0 from SNR? In your codes you have written Eb_N0_dB, is this EbN0?? I think it is SNR. If this is EbN0, why am i not getting same curve for BPSK and QPSK… Please Help

  102. i have a question..
    i have a program wich makes bpsk modulation and i need to show Bit error curve for BPSK modulation – theory, simulation (both) as you do here in this page at top.

    ok but there is something that happens with simulation curve that appears a little bit more to the right side of teoric curve.

    i wanna know a reason why does it happens.

    1. @Eduardo Solano: If the simulation curve is happening to the right of the theoretical curve, it typically means that you are adding more noise than what is required. Also make sure that you simulate for atleast 10^6 bits, such that you get statistically accurate results.

  103. hi every body can any one tell me the code for “signal to noise ratio using autocorrelation in time domain” using MATLAB

    thanks in advance…..

  104. thnx for ur response,,it means dat u need to have atleast 2 no of rcvd bits at da reciever 2 be compared..but my confusion is that how to use vblast when i am considering diversity mimo,where i have to use combing technique first?? can i implement vblast in dat case or not???

    1. @maya: Well, try to write the equation in matrix form. For eg, consider a simple case where we have two transmit antennas and 3 receive antennas. The matrix dimensions will be of:
      y = Hx + n, where
      y – received symbols of dimension [nrx x 1]
      H – channel matrix of dimension [nrx x ntx]
      x – transmit symbols of dimension [ntx x 1]
      n – noise of dimension [nrx x 1]

      If we write the zero forcing equalizer for this equation, then we can find that
      W = (H^H*H)-1*H^H

      This equalization also performs the diversity combining via Maximal Ratio Combining way. Does this help?

  105. HEY sir can u plzz help me in getting the main concept of vblast detection scheme,,i am not geting how it estimates the data…

    1. @maya: Well, understanding V-BLAST with Zero Forcing equalization is the most simplest way. From our linear algebra days, we know that if we have two unknowns, we need atleast two equations to solve them.

      Adapting them to MIMO V-BLAST, the two unknowns are the unknown transmitted data, and the two equations are provided by the two receive antennas. Just putting a simple example,
      y1 = a*x1 +b*x2
      y2 = c*x1 + d*x2
      y1, y2 are received symbols
      a,b,c,d are channel taps (which are known) and
      x1, x2 are unknown transmit symbols.

      The following post might be of bit more help.

      Hope this helps.

  106. Dear krishna
    I hope you are fine. I am in 5th semister of my bachelors of telecommunications. So I am new to this field and facing lots of difficulties. I find your site very impressive and have joined it recently. Can you kindly tell me why we normalize things? thnx

    1. @aam_log: Good luck for your studies. We normalize, because we want to have a fair comparison when trying out different approaches….

      For eg, if we are comparing 100apples with 100grapes, we would want to know the price per kilogram, no?

  107. hello sir

    i am doing my project in cdma can u tell me the purpose of rake receiver and details about maximum ratio combining

    1. @kanchana: Well, I have not quite discussed about cdma in the blog till date. Hope to do so in future.
      I beleive the intent of the rake receiver is to coherently combine information from all the multipath taps such that the bit-error rate is reduced. I have posted about maximal ratio combining for a 1 transmit 2 receive receive diversity case.
      Though the context is different, I think you should be able to adapt that to suit your needs.

      Good luck.

  108. wah ini dia data yang aku cari-cari akhirnya menemukan yang namanya BPSK
    thanks yaaaaaaaaaaaaaaaa…
    sip sip sip sip

  109. Dear Krishna Pillai
    Thanks a lot for ur suggestion.When I domodulate following code it gives a simple sine wave.How can I demodulate it?
    close all
    b=[0 1 1 0 1 0 1 0 0 1];
    while n<=length(b)
    if b(n)==0 %0 is -1
    plot(tx,m,’LineWidth’,1.5);grid on;
    plot(tx,m,’LineWidth’,1.5);grid on;
    hold on;
    title(‘BPSK modulated Signal’)

    How coherent detection can be applied on it?
    Thanks in advance.


    1. @Student: hmm..let me try to understand. For bit0 you are sending -ve sine wave, and for bit1 you are sending +ve sinewave right. At the receiver you cmay undo the effect of the sine wave by multiplying the received signal with the sine wave and taking the mean. Then perform hard decision decoding on that signal.
      Does that help?

  110. Hi,
    I would need equations of bit error probability for this modulation technique: MSK, GMSK, QPSK, ASK. Do you know it?
    Thank for answer

  111. Dear Krishna Pillai
    How can I correlate demodulated signal and binary code in BPSK ?I mean reteriving back of binary code.

    1. @Student: At the receiver, using hard decision decoding you can find out whether the received symbol is +1 or -1. Then we can easily map to +1 to bit1 and -1 to bit0 respectively.

  112. Please i dont know how i can acess the e book.
    I am working on super orthogonal space time turbo codes in OFDM systems and will need this e book .Thank u

  113. Dear Krishna Pillai :
    How can I change Amplitude Modulation to BPSK(y1 = ammod(x1,Fc1,Fs) to bpsk . I hope for a positive response.
    close all
    clear all

    t = 0:0.00001:0.001;

    Fc1 = 1000;

    Fs = 12000;
    y1 = 1;
    x1 = cos(2*pi*1000*t);

    in_p = input(‘\nDo you want to enter user Y/N: ‘,’s’);

    if(in_p == ‘Y’ | in_p == ‘y’)
    y1 = ammod(x1,Fc1,Fs);

  114. Hello,

    Do you know how you can build a zero-crossing detector or delay-and-multiply detector for the FSK.


  115. Hi, can you help with this issue,consider QPSK transmission over flat fading Rayleigh channels and a 1×2 system.The channels on the different diversity branches are assumed to have the same variance and the noise samples are assumed to be uncorrelated and circularly symmetric Gaussian variables with the same variance.
    Consider three combining schemes: MRC, EGC and SC, for plot the BER versus the average signal-to-noise ratio (SNR).
    I hope you can help me

  116. Hi Krishna,

    All your posts have been very helpful and easily understandable. I was looking for a code that has all the modulations(bpsk,qpsk,mpsk,ask,bfsk,msk etc…) in a single program and to compare their BER, PSD curves. please let me know if you have any.


  117. Can someone pls provide me the matlab code for the below mentioned project work:

    A randomly generated bit stream should be transmitted through an AWGN channel using BPSK. THe received symbols should then be converted again to a bit stream. The received bit stream should be compared with the original one to determine the total number of errors and hence the bit error probability. The probability of error should be calculated for several values of EB/No.
    Once the above system is succesfully simulated an additional block of channel coding should be incorporated in the system. A (15,11) Hamming code should be used for this purpose. Simulation of this should be performed to increase the performance of the system.


    1. Bit error probability versus EB/No graph of BPSK
    2. Bit error probability versus Eb/No graph of BPSK using equation
    3.Bit error probability versus Eb/No graph of BPSK system with channel coding/decoding

    The written code shuld allow the user to perform simulation for any number of bits and for any range of Eb/No.


    Looking forward for your assitance.

  118. hi
    How can i calculate throughtput for a multi-hop wireless network using matlab. please give me some suggestion ragarding this….


  119. Thanks for your codes.It’s helped me a lot.But I want to replace Eb/N0 as SNR how can I do it.Could you pls help me urgently.

    Thanking you

  120. Hi there, very useful site. I have a couple of questions regarding the BPSK:

    1) Could you go in to a little more detail why you use complex noise vs. real. Although it doesn’t make a difference for BPSK, I’m assuming it actually will with other modulation methods. I didn’t see this in any of my textbooks.

    2) In your code for the BPSK, you divide by 20. Shouldn’t that be 10? Given an SNR in dB, we get S/N0 = 10*log(SNR-dB / 10).
    This gives N0 for a given signal power S.
    The variance^2 = N0/2 and if we assume 0 mean, this gives us our AWGN, which is a random variable describing the noise amplitude. In my simulation, if I use 10 instead of 20, my results were way off, but 10 gives a perfect match with theory.

    Thanks much,

    1. @ Matt K: My replies
      1). Yes, for BPSK the noise component on the imaginary arm is ignored. To make the noise model same as with other modulation schemes, I kept it as complex.
      2). The division by 20 is to convert the dB to scale the voltage signal. In my simulation model, the noise term n is a Gaussian random variable with mean 0 and variance 1. Therefore, I am scaling the noise voltage by 10^(-Eb_N0_dB(ii)/20)*n. Thats why I used 20.
      In general, it depends on how one creates the model. Different people have different coding style. However, its the end result which matters. 🙂

      Hope this helps.

  121. tq krishna…i understand it more than before..i really appreciate it..
    however can i ask u bout the graph of BPSK between simulation and theoretical in term of BER vs SNR…
    how the shape of graph become like that?

  122. Hey krishna,

    I am new here. I find this site really very useful. Thanks for everything. I have a question reg BPSK sim. I did it with awgn function and I am not getting theo and sim graphs overlapping. Please answer this as soon as you get time . thanks

  123. krishna can u help to understand more about the function of transmitter filter in BPSK and also how the ISI(intersymbol interference)will occur at transmitter filter??

    1. @nyna: In the transmitter, we will first be converting bits into analog voltage (+1/-1 in the case of BPSK). However, we cannot transmit them as is, as it will occupy lots of spectrum (which is not desirable). Hence we need to do filtering. However, when doing filtering, we introduce ISI (inter symbol inteference) i.e each symbol may interfere with the next symbol and so on.

      So, the question is: can we do filtering without introducing ISI? The answer is YES. There are simple filters like rectangular, ideal filter like sinc and practical filters like raised cosine, which can do the job. You may read more about them @

      Hope this helps.

  124. hai..krishna..i want ask you the function of hamming code and bCh because i already simulate the block using that coding…can you give me the formula for i’m calculate theory the bit error rate when using coding for i compare between theory and simulate…..also can you give me the graph ber vs Eb/No when coding..tq

      1. sir,
        i want matlab programs for MCCDMA,whtever u hve on dis topic ,plz mail me as soon as possible,since MCCDMA is my thesis topic, but i m facing many prblms regarding matlab programs

        thank u,

  125. The term 10^(-Eb_N0_dB(ii)/20) is for scaling the noise power as per the defined bit to noise ratio, why divided it by 20 >>??to convert it to voltage

  126. hi i have plotted the results for some coding
    i just want is biterr with no coding

    i.e. simply take msg, modulate add noise and demodulate

    now find the biterr or symerr for oiginal msg and recovered msg

    pls help as early as possible

  127. hi … i think my username “invizible soul” has really made me invizible 🙂 … as no one has answered my questions …

  128. hello..
    who will help me to understand the function of a filter in a block diagram of a basic communication system in BPSK?
    What the purpose of finding the value of delay in the program of BPSK?

    1. @nyna: Filter is used for controlling the spectrum of the transmission. However, when doing the filtering we do not want to introduce ISI. Hence we typically go for filters like raised cosine etc. You may find some more details @

      Hope this helps.

  129. kindly reply me !!! i am still waiting … from where we get this expression:


  130. thanks khrishna with your help………..

    i late reply n visit your site coz im busy….
    how abaut this???
    % BER_sic=sim_sic_fn(SNR_dB, P, Nb, code_matrix, chan_type)
    % return bit error rate of the sic receiver in AWGN or rayleigh fading
    % SNR_dB=signal-ti-nopise ratio in dB
    % P=power control vector, P(i)=transmitted power of i-th user
    % Nb=number of transmitted bit
    % code_matrix=matrix of the spreading code used
    % chan_type=channel type(1=AWGN, 2=rayleigh fading channel)
    % output:
    % BER_sic=bit error of the sic receiver
    function BER_sic=sim_sic_fn(SNR_dB, P, Nb, code_matrix, chan_type)

    K=length(P); %number of user
    N=size(code_matrix,2); %spreading factor


  131. hi hope you are fine. In your code you have scaled the noise with the term 10^(-Eb_N0_dB(ii)/20)
    my question is why we do this?I mean why we need to scale the noise power as per the defined bit to noise ratio? Any references will help me a lot
    fire back soon plz !!!

    1. @invizible soul: Sorry for the delayed response. I am traveling with out a reliable access to internet.

      In this analysis, our objective is to analyze the impact of noise on the reliable decoding of BPSK modulation. To analyze the impact, we need to find out the performance (bit error rate) for different values of noise. Hence we scale the noise by noise power.

      Hope this helps.

    2. @manju I got a better solution for your question.We need to untilmately multiply with standard deviation of noise so we get 20 there.

      1. SNRdB = 20 log (Asignal / Anoise)
        -SNRdB = 20 log(Anoise / Asignal)
        -SNRdB/20 = log(Anoise / Asignal)
        10^(-SNRdB/20) = Anoise/Asignal <——— noise amplitude scaling factor.

  132. Hi krishna

    can any one help me for code SIC (Successive Interference Cancellation).i will tray it for study…..


    1. @manju: You may use the following steps provided in the Matlab code snippet
      clear all
      N = 7; % number of bits
      ipBit = rand(1,N) > 0.5; % random 1’s and 0’s
      ipMod = 2*ipBit – 1; % converting 1’s to 1 and 0’s to -1
      os = 4; % oversampling factor
      ipMod_os = [ipMod; zeros(os-1, length(ipMod))]
      ipMod_os = ipMod_os(:).’;
      ipMod_filter = conv(ipMod_os,ones(1,os)/os);

      Does that help?

  133. Hi all

    hope u all r f9

    can any one help me for BER performance of adaptive modulation
    (QPSK,4QAM,16QAM)in AWGN only I need to draw them in single graph

    thankz in advance

    1. @mak: As I understand from your query, you want to switch from one modulation scheme to another based on a defined error rate constraint. For eg, for lower SNR’ you would want to use BPSK, then move on to QPSK/4QAM for medium SNR’s and then hit 16QAM at high SNR’s.

      I do not have any code explicitly doing this, but I would think that it would be reasonably easy to build using the following posts:
      (a) Bit Error Rate (BER) for BPSK modulation
      (b) Symbol Error Rate (SER) for QPSK (4-QAM) modulation
      (c) Symbol Error Rate (SER) for 16-QAM
      (d) 16QAM Bit Error Rate (BER) with Gray mapping

      Hope this helps.

  134. Hi Krishna,
    Thanks for the good information.

    Could you shade light on the derivation of Sigma^2 = No/2 from correlation and spectral density of noise stand point?


  135. @Ideal: It just means that any value of y (real component) greater than 0 is assigned to 1 and if less than 0 is assigned as 0. This is called hard decision decoding.

    Soft decisions – where we do not decide on what was transmitted based on the received constellation – are used typically when there is a decoder block (like Viterbi) following the constellation demapping.

    Does this help?

  136. hi
    Dear Krishna,

    % receiver – hard decision decoding
    ipHat = real(y)>0;
    here what do you mean by hard decision decoding? plz reply in brief.

  137. @Ideal: So you are able to plot both the curves using independent functions/scripts, but not able to combine them. Correct? This should be a reasonably easy aspect to resolve.
    You can have a main script which generates the transmit symbol, Eb/No values etc. Pass it two function
    (a) for awgn channel y = x + n,
    (b) for rayleigh channel y = hx+n
    Each function will count equalize, count the errors and report the number of errors for each value of Eb/No.

    That should address your concern. Hope this helps.

  138. Dear Krishna,

    I have made a program for BER of BPSK in AWGN and Rayleigh fading channel, now I want to combine the both results in one graph, when I combine the results, the curve for rayleigh change but for awgn not change, can you give any idea what should i do, either i make a functions of both separately and call in one program or i should do something else?

    anyone have idea about it plz comment

  139. for QPSK modulation, the variance is

    sigma = 1/sqrt(10.0^(snr/10.0));

    Do you have any idea why it has to devide by 2?

    Thanks alot

  140. @grad student: Yeah, thanks for the suggestion. I agree that your proposal will make the text even more readable.

    Anyhow, may I recommend to keep status quo given that the current text is not making it very difficult for the reader.

  141. Nice graphics and derivation.

    Might want to consider using a separate integration variable for the definition of the erfc function, since the variable x was used for both the erfc argument and integration variable.

    Perhaps just use ‘z’ or ‘y’ for the erfc integration variable, i.e., erfc(x) = 2/sqrt(pi) * Integral_x_inf { exp(-z^2/2)*dz } . This will then make it clear that the erfc argument ‘x’ is used to modify the integration limits, as opposed to the actual function being integrated.

  142. @Hussein: Thanks. I do not have the codes, but I would think that it will be reasonable to extend the available simulation models to 16QAM case.
    For 16-QAM in AWGN (without OFDM) you may look at the posts,

    For extending them to OFDM, you may use the following posts as reference:

    Hope this helps.

    1. hi mr krishna
      i want your mail becauce i have many problem about matlab programming
      plz send to me your mail

  143. Hi Krishna

    thank you very much for Your simulation and theory prgram, really they are very useful. Do you have the matlab code for OFDM with 16-QAM in AWGN and Rayleigh Channels.
    Best Regards

  144. Hello Krishna

    Your simulation and theory prgram is very useful. Do you have the matlab code with qpsk symbols ? Would be great !

    I want to compare the theoretical bit error curve for qpsk with a simulated bit error curve with a zero force equalizer. Can you please give me an advice of how to implement this.

    Thank you in advance

    Best regards


  145. @farie: The demodulation approaches for PSK is a different from QAM. For 16-PSK case, you can find two posts:

    Symbol Error Rate for 16PSK

    Bit error rate for 16PSK modulation using Gray mapping

    You should be able to modify the code for 16-PSK to 8-PSK case.

    For 16-QAM case, there are two posts on error rate computation:

    16QAM Bit Error Rate (BER) with Gray mapping

    Symbol Error Rate (SER) for 16-QAM

    Hope this helps.

  146. @ravi kumar: In AWGN, there should not be any advantage by using OFDM modulation. OFDM can be thought of as a simple up-conversion scheme where each information gets multiplied on different sub-carrier frequencies.

    However, the BER difference should come in multipath simulations.

  147. hi,

    Is there a demodulation matlab code which i can apply on most modulation methods like 8 psk, QPSK, 16QAM..etc.

    Also do you have 8psk and 16 QAM modulation and demodulation codes in matlab?

    thank you

  148. hi,
    I do not understand why the BER curves for both BPSK and BPSK with OFDM are almost same where the BPSK with OFDM get
    advantage i.e., decrease in BER in case of BPSK with OFDM.
    So please give me the idea by differentiate the two curves and how the BER can be decreased with OFDM as soon as possible. iam waiting for u r response hopefully,thanku

  149. @antiwina: You are right, the imaginary component is ignored. Infact if you see the simulation model, for demodulating the symbol, we use only the real part of the received signal.

    However, our definition of noise is with a variance of N0/2 on the real arm and N0/2 on the imaginary arm.

  150. Hi,

    I do not understand why a complex gaussian noise is required. The source here belongs to a BPSK modulation so I think, in practice we have a real noise, because we do just activate the in phase component and not the quadrature phase component.

  151. thanks a lot code is very useful

    even i want a code using upsampling and pulse sampling in BPSK mod and downsampling and pulse sampling in BPSK demod

  152. @Jhony: If I may put very briefly, fading is the characteristic of the wireless channel. The channel (read enviornment) will vary with time and will exhibit different behaviour for differnt frequencies.

    One of the most simplest and useful model of fading is to visualize it as filter whose coefficients change in time.

    Ofcourse, this is a simplistic description. For a detailed overview, may I recommend chapter 2 from the book
    Fundamentals of Wireless Communication – David Tse, Pramod Viswanath

  153. hi i hope you are in good health. Can you please tell me what a fading co-efficient is? and what does it signifies in few lines. I shall be very thankful to you.

  154. thanks alot … u helped me alot i m very happy to learn all this … thanks once again … and plz keep helping me as i want to learn …

  155. [randn(1,N) + j*randn(1,N)] generates complex Gaussian noise with mean 0 and variance 2. To make the variance to 1, the scaling factor of 1/sqrt(2) is used.

    yes, the j term is for complex noise.

  156. 1/sqrt(2)*[randn(1,N) + j*randn(1,N)];

    1/sqrt(2) ==> this is the scaling term?
    j*randn(1,N) ==> this is to generate complex noise values?
    I am thankful to u for helping me

  157. @Jhony: The randn() function generates a random variable having Gaussian distribution and the spectrum of this random variable is same across frequencies. Hence the name White Gaussian Noise (WGN).

    This is then Added to the signal, hence the name Additive White Gaussian Noise (AWGN)….

  158. yeah ! trying to get hold of things slowly now… i hope u dnt mind me asking foolish and lots of Qz…
    about the formula u uzed for awgn … i didnt get how it produces awgn … 🙁

  159. @Jhony: Thanks. Am doing good. How are u? Glad to notice that you are not desperate now 🙂
    Typically, to keep the code simple, I tried to avoid in-built functions. The randn() function + scaling the does the job. Thats all.

    1. hi krishna pillai i am doing project on viterbi decoder i want the matlab code for the viterbi decoder if u have the code please send me

  160. hi
    hope you are doing fine…in the matlab program u have given here, why didnt you use the awgn matlab function to introduce the noise? why have we taken this formula? thanks

  161. @babar:
    The term 10^(-Eb_N0_dB(ii)/20) is for scaling the noise power as per the defined bit to noise ratio. The for loop is for computing the bit error rate for each Eb/N0 value.

    1. Hi..
      Why you are multiplying ‘n’ with The term 10^(-Eb_N0_dB(ii)/20).
      The didnot understans the concept behind this from the previous posts..

      1. @srinivas: This is to scale the noise term ‘n’, so that we can obtain BER for different values of Eb/No. In the simulations, we define Eb/No in dB. To convert dB to define a scaling term for noise voltage, we use 10^(Eb_N0_dB/20).

        Hope this helps.

  162. hello! i am unable to understand the FOR loop operation in this code kindly explain a bit
    thnaks in advance

  163. sorry for my previuos question i got it just after posting my previuos mesg…its coz we want to add the noise to the signal so we need to convert it to dB … right? thnx

  164. why are we using this term 10^(-Eb_N0_dB(ii)/20) in the code to get the final signal y. why cant we write y=s+n; ???????????????

  165. @ David: Thanks 🙂

    Oh… were you unable to find the Matlab/Octave code? The link to the code is under the heading Simulation model.

    Anyways, for BPSK it is simple. We observe the real part of the received symbol. If the real part is greater than 0 decode as +1, if the real part is less than 0 decode as -1.

    Once we have the decoded symbols available, the number of differing symbols can be obtained by
    nErr = size(find([ip- ipHat]),2);
    Ofcourse, using find() is only one among the many different ways to compute this.


  166. Hey,
    I found theBit error curve for BPSK modulation – theory, simulation very useful. How do we demodulate , decode and compare it with the input signal

    1. thanks alot or this article . it really helped me alot .
      and I am waiting the ofdm article.

    2. my project is to design a digital communication system using channel coding.

      help me plz…
      if u have matlab code and using simulink model please mail me..

  167. Dear Krishna,
    Your simulation and theory prgram is very usefull and helpfull.
    Could you please send me the syntax matlab comand of QPSK modulation for OFDM under multipath (i.e 3 path) fading.
    Very thanks in advance.

  168. Thanks.
    I have not checked for FSK modulation. Once I understand, hopefully will be try put together a post.

  169. Hey your BPSK theory and simulation is very useful.Do you have it for FSK modulation too?If you don’t mind please send to me. thanks.

    1. close all
      clear all

      N = 10^6 % number of bits or symbols
      rand(‘state’,100); % initializing the rand() function
      randn(‘state’,200); % initializing the randn() function

      % Transmitter
      ip = rand(1,N)>0.5; % generating 0,1 with equal probability
      s = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 0
      n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % white gaussian noise, 0dB variance
      Eb_N0_dB = [-3:10]; % multiple Eb/N0 values

      for ii = 1:length(Eb_N0_dB)
      % Noise addition
      y = s + 10^(-Eb_N0_dB(ii)/20)*n; % additive white gaussian noise

      % receiver – hard decision decoding
      ipHat = real(y)>0;

      % counting the errors
      nErr(ii) = size(find([ip- ipHat]),2);


      simBer = nErr/N; % simulated ber
      theoryBer = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))); % theoretical ber
      theoryBer1 = 0.5*erfc(sqrt(0.5.*(10.^(Eb_N0_dB/10)))); % theoretical ber

      hold on
      axis([-3 10 10^-5 0.5])
      grid on
      xlabel(‘Eb/No, dB’);
      ylabel(‘Bit Error Rate’);
      title(‘Bit error probability curve for BPSK modulation’);

      hold on
      axis([-3 10 10^-5 0.5])
      grid on
      xlabel(‘Eb/No, dB’);
      ylabel(‘Bit Error Rate’);
      title(‘Bit error probability curve for BFSK modulation’);

      In bfsk need double bit error rate to maintain the same avarage error rate in BPSK.
      Feeel the difference..

        1. Hey Krishna,

          I need help regarding the simulation for the fig 1 for the IEEE paper:
          Xiaodi Zhang and N.C. Beaulieu, “A Closed-Form BER Expression for BPSK Using MRC in Correlated CCI and Rayleigh Fading,” IEEE Trans. Communications, vol. 55, no. 12, pp. 2249-2252, Dec. 2007.

          I am getting stuck with the simulation code. Can you please give a possible MATLAB code.


        2. I want to change the spatial resolution in dots per inches of an image.
          Do u have any code regarding this problem?
          if so plz send me

    2. One who is interested in including his/her name in 3 research conferences in international IEEE conference should reply on email address Both papers are related to mobile communication. Total of 3 authors list will be included in each Paper. 2 author names have already been included. Interested candidates who want to include his/her name at 3rd position will be required to pay for the registration fee.
      Matlab Code as well as the full paper will be sent to the individual after acceptance of paper from the conference.
      Fee submission will be through Freelancer and elance. Candidate name will be included in Paper after milestone payment is released by him/her. Milestone payment is one which is in the custody of broker (Freelancer or Elance authority) and not in the custody of either party. After the paper is accepted and the client is conformed about the acheivement, only then he will be allowed to pay. Time is short so the policy of 1st come 1st serve will be entertained.

Leave a Reply

Your email address will not be published. Required fields are marked *