We have quite a few articles discussing bit and symbol error rates for popular digital modulation schemes in **A**dditive **W**hite **G**aussian **N**oise (**AWGN**) channel. This post summarizes the articles discussing the **theoretical** and **simulated** **error rates** for the digital modulation schemes like **BPSK**, **QPSK**, **4**–**PAM**, **16PSK and** **16QAM**. Further, Bit Error Rate with **Gray coded** mapping, bit error rate for BPSK over **OFDM** are also discussed.

The links to the individual articles and the **Matlab**/**Octave** **simulation models** are listed below. Alternatively, I have made a **e-book** discussing all the below mentioned articles to a single PDF file. If you wish, you can download the **free e-book** by subscribing to the **free email newsletter**.

## Subscribe and download the free e-Book

**Note:**

Existing subscribers can contact me to access the e-book.

The links to articles discussing and the Matlab/Octave simulation models are listed below.

## Bit error probability for BPSK modulation

.

Click here for Matlab simulation for bit error rate (BER) curve with BPSK modulation.

## Symbol error rate for 4PAM

.

Click here for Matlab simulation for symbol error probability with 4PAM modulation

## Symbol error rate for 4-QAM (QPSK)

Click here for Matlab simulation for symbol error probability with 4QAM (QPSK) modulation

## Symbol error rate for 16-QAM

Click here for Matlab simulation of symbol error rate curve with 16QAM modulation

## Symbol Error Rate for 16PSK

.

Click here for Matlab simulation for symbol error rate with 16PSK modulation

Note: The formula derived in the post is for a general M-PSK case. For an M-PSK scheme, the symbol error rate is,

.

## Subscribe and download the free e-Book

**16QAM Bit Error Rate (BER) with Gray mapping**

Click here for Matlab simulation for bit error rate for 16QAM modulation with Gray mapping

## BPSK BER with OFDM modulation

Click here for Matlab simulation for BER with OFDM modulation

**References**

[DIGITAL COMMUNICATION: PROAKIS] Digital Communications by John Proakis

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

[COMMUNICATION SYSTEMS: PROAKIS, SALEHI] Fundamentals of Communication Systems, by John G. Proakis, Masoud Salehi

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Please help me to understand my results.

I have simulated in Simulink (Bernoulli, Hamming Encoder, BPSK Modulator, AWGN, BPSK Demodulator, Hamming Decoder) BPSK with AWGN and have obtained that Hamming (7,4) code has lower BER than Hamming (31,26).

Is it connected with noncoherent demodulation?

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@Grekhov: Replies

1. Hmm.. where you comparing Eb/N0? Would expect that you are using coherent demodulation

Did you want to look at https://dsplog.com/tag/hamming/

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hi, can you please tell me what are the changes to be made in the symbol error probability code for QPSK if I want the bit error rate instead?

The question is- bit error probability for QPSK

@Bharg: For QPSK, one needs to send information on both real and imaginary components

https://dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/

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and THANK YOU for the cool work!

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BTW this site is really really what i need !!!

continue the great work

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sir i require qpsk mod/demod using pn sequence in cdma environment

@lakshmi narayana: Sorry, I have not discussed CDMA scenarios in the blog. However, I would guess it will be reasonably simple to adapt the QPSK mod/demod post to the CDMA case.

https://dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/

At the transmitter, multiply each symbol by the PN chips. Correspondingly, at the receiver convolve with the PN chip and pick the maxima.

Dear Krishna Sankar

I have paper (journal ) but I dont know simulate it, pleaze help me to send me the code to summerize it.(this is homework)

Peak Power Reduction of

OFDM Signals with Sign Adjustment

Masoud Sharif, Vahid Tarokh, and Babak

thank you

@MAH: Sorry, I wont be able to code for you. However, you can ask queries in the comments section. I will try answer to the best of my knowledge

Dear Pillai

Ineed Derivation of FSK BER in Rayleigh channel

and Ihave paper (journal ) but I dont know simulate it, pleaze help me to send me the code to summerize it.

Peak Power Reduction of

OFDM Signals with Sign Adjustment

Masoud Sharif, Vahid Tarokh, and Babak

thank you

@MAH: I have derived BPSK BER in Rayleigh channel @

https://dsplog.com/2008/08/10/ber-bpsk-rayleigh-channel/

https://dsplog.com/2009/01/22/derivation-ber-rayleigh-channel/

I am not sure whether FSK results will be 3dB poorer that the BPSK case, but I would guess so.

Sorry, I do not have simulation for the mentioned paper.

can u send me a copy of the pdf file, i just registered

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sir, please send me QPSK based matlab code with alamouti code and decode facility

@shreedhar A Joshi: I posted BPSK based Alamouti coding and decoding https://dsplog.com/tag/alamouti

It should be easy to adapt to QPSK case

I’ve learned a lot from your articals,thanks for your work!

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Hi,

I have already ask you the question about the calculation of uncoded BER for the pulse position modulation and ergodic capacity calculation using any fading channel(lognormal or gamma). please reply

@Amjad: Sorry, I need more study to reply to your queries. And due to time constraints, am unable to allocate time. Kindly excuse.

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thank you for sharing such information.it is helpful in acquiring the knowledge about programminig in MATLAB.

can u help me in the topic use of wavelet transform in OFDM .

i m doing a project on it.

@gul: Sorry, am not familiar with wavelet transform in OFDM

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hi!

we are doing a project to implement adaptive channel equalizer using MAT LAB.

can you provide us coding of adaptive channel equalizer.

@ahishek: I have not yet posted articles on adaptive equalizer. good luck.

hi im student of final year telecommunication

i need the matlab simulation codes of the following data

mobile wimax(error detection,correction,modulation)

same for hsdpa

@aisha: Sorry, I have not discussed specific to 802.16e. However, you may find posts on QAM and Viterbi @

https://dsplog.com/tag/qam/

https://dsplog.com/tag/viterbi/

Hi,

I have subscribed for the e-book, but i haven’t received any. Can you tell me the procedure to download it?

@ReiRok: I emailed you the download instructions. Sorry for the delay in my reply

I need equations on QPSK so that i can calculate bit error and eye diagram using matlab. Can u help me?

@rasi: Please refer to the posts

https://dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/

https://dsplog.com/2008/05/01/eye-diagram-plot-matlab-raised-cosine-filter/

Hi Krishna,

Your blog helps me to learn OFDM , thank you very much for that.

Could you send me the download instructions ?

I am try to simulate BER for OFDM using 16QAM

I would appreciate your help.

Thanks!

@Ryan: You may refer to the posts:

https://dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/

https://dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/

https://dsplog.com/2008/06/10/ofdm-bpsk-bit-error/

https://dsplog.com/2008/08/26/ofdm-rayleigh-channel-ber-bpsk/

Good luck.

Hi Krishna

I would like to download your ebook on AWGN channels

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Pit

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please tell about demodulation techniques of single side band suppressed carrier within two days. please reply within two days.please……

@vanitha: Sorry, the two days have long passed 🙂 Hope you found the required information.

Hi,

Could you give a copy of free ebook ?

You really set up a good website of communication system.

Very appreciate your effort!

Thanks.

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AOA

Dear friend!

i am doing thesis of ms, my topic is

Channel State Information Enhanced Adaptive MC CDMA.

can u people help me, specially in matlab coding………

@Tahir Abbas: Good luck for your thesis.

hi , i hav tried qpsk with half rate convolutional coding with trellis structure of t = poly2trellis(7,[171 133]); i have also applied it with and without gray coding but still i m getting ber of 0.5.. why it does not match theoritical ber theortical ber……plzzz can any body tell me where i m wrong ….i had also tried 2/3 rate coding it works fine but ber at lower EbNo values is little bit higher after that remaining ber values are ok..but i m more intereted in 1/2 rate coding

any help will be appreciated

thanks in advance…

hi krishna,

i want to plot different modulation schemes BER vs Es/No in awgn channel on a single plot.for example, 0 to 9 db QPSK modulation, 10 to 11 db 16 QAM modulatio,12 to 19 db 32PSK modulation and so on. its output will be a single line at series of Es/No values.can u help me about this?

thnx

@farrakh: From your question, it sound like you want to build an adaptive modulation simulation. Though I do not have posts discussing this explicitly, hope you will be able to adapt the QPSK, 16QAM and 16PSK simulations to meet your need.

https://dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/

https://dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/

https://dsplog.com/2008/03/18/symbol-error-rate-for-16psk/

Hi Krishna, I am working with mobile location in IS-95 CDMA system. Could you please send me matlab simulation m-file for QPSK modulation and demodulation of Pilot signal in IS-95 CDMA for Pilot channel acquisition. It would be a great help in my research.

Best wishes,

Kabir

@Kabir: Sorry Kabir. I have not tried modeling IS-95 systems.

hi,

i am working for bit error rate for BPSK, QPSK and 16 QAM modulation techniques for different code rate 1/2, 3/4, 9/16 using AWGN and Rayleigh channel so plz help me for changing code rate and delay in hiperlan/2

@Girish: How do you achieve different code rate – by puncturing the convolutional code?

its good

I need to build a code of probability of Bit Error rate of BPSK, 16 QAM and 256 QAM and QPSK. I need some help. where could I get help . Can you Please help me on that regards.

@Gazi: You may use the following links

a) Bit error probability for BPSK modulation

https://dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/

b) Symbol error rate for 4-QAM (QPSK)

https://dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/

c) 16QAM Bit Error Rate (BER) with Gray mapping

https://dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/

Hi Krishna,

I would really appreciate if you can help me to implement the cooperative communication (amplify and forward ) and (decode and forward) over the slowe fading channel . the system consistes of source, relay and destnation

all the best

@taher: Sorry, I have not tried modeling co-operative communications.

Dear Taher

I am also worikng on multi-hop relay network. I am new in this area and need some help from you. if you can give me ur email address please.

my email is nguyenkimhieuha@gmai.com

i’ll wait for ur reply,plz contzct me on my email.

Hi Krishna,

Please send me your e-book on AWGN.

Thanks!

-John

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please send me ebook for BER in AWGN

thanks

regaurds Girish K.

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Hi, I’d like a copy of the e-book as well.

Thanks

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Hi Krishna,

I would really appreciate if you could send me the ebook.

Thanks,

Carlos

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I am a member; can you please send me the URL for the ebooks you have? Thanks a lot.

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I need errors in computing like inherent errors,absolute, relative, modeling errors…..

please send on my mail alimabrook@gmail.com

with subject as ERRORS

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hello sir,

am yet to get the ebook.

I would appreciate it ,if you send me the link

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sir

plz mail me link to download book…..

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Plz send me the ebook on Error Prob

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Dear Mr.Krishnapillai,

I admire your dedication and sincerity in helping others by giving useful suggestions, ample guidance and valuable advises. I am doing research in communication field and topic is to improve the performance of ofdm systems. my plan is to minimise PAPR and Frequency offset by using scrambling(PAPR Redn) in Txr and Windowing in Rxr(ICI Cancellation), plz comment on this and if you have any suggestion it will defenitely help me.

@Bijunair: Well, even I have not done much on PAPR reduction rather than using scrambling at the transmitter. Am curious about how you are going to reduce the effect of frequency offset by using windowing in receiver. You may point me to some references which you are using. Thanks.

hi Krishna

how can i get this book? can u help me and send the downlaod link? thanks alot

@radman: I just mailed you the download instructions.

hello sir

i am already subscribed but i did’nt get that ebook

@kanchana: If you had subscribed, you should have got the email download instructions. Maybe the email got incorrectly flagged as spam. Anyhow, I just mailed you with a fresh set of intructions.

Hai, krishna ,

Please send some information about Space time block codes.i.e introduction with two pages for the documentation.

Please…please send

by

madhu

@madhu: You may find two articles discussing Alamouti Space Time Block Codes @

https://dsplog.com/tag/stbc/

Hope this helps.

pls kindly send me the e book. thanks

Hi, Krishna , i have subscribed the newslater, please send me the download link! Thanks! Expecting your reply!

@Danial: I just mailed you the instructions. Please check.

Sorry, I didn’t received your e-mail yet.

Would you please resend the e-mail to me?

@Danial: Re-send

Hi,

I have subscribed for the e-book, but i haven’t received any. Can you tell me the procedure to download it?

Thanks

@Murali: I mailed you the download instructions. Please check.

Hello,

You are really doing a great job!

pls send me the link to the ebook

thanks alot

@val: I send you the download instructions over email. Please check.

Please send me the link for the ebook, thanks a lot.

@wig: I had mailed you the instructions. Please check

I too need this e-book. Please send the same

@Munish: I had mailed you the download instructions over email. Please check.

Please send me the link for the ebook, thanks!

@coss: I had mailed you the download instructions. Please check.

i am already a member pls send me URl i want this book

@mansi: I had mailed you the download instructions over email. Please check.

Krishna plz send the above mentioned book. Thanks

@Muhammad : I had mailed you the download instructions. Please check

hi

kindly mail me ebook

regards

pulkit

@pulkit: I just send you the download instructions over email

hi

kindly mail me ebook

regards

sara

@saraviqar: I just send you over email the download instructions

please suggect me about EFFECT OF MULTIPLE ACCESS INTERFERENCE IN W-CDMA UPLINK design project (or)any material

@navya: On a high level, effect of multiple access inteference is equivalent to a MIMO communication (V-BLAST) where each dimension interfers with the other. You may read more about MIMO with Zero Forcing equalizer @

https://dsplog.com/2008/10/24/mimo-zero-forcing/

Hope this helps.

Hi Krishna,

How do I get a copy of the ebook, since I am already a member ?

@Arif: I mailed you the link.

hw to download the boks

@mahesh: I have mailed you the download link

Hi

Would like a copy of e-book onerror prob.

Thanks

@A-Imam: I mailed you the download link.

3ks

Hi, could you please send me the copy of the ebook Fundamentals of communication systems by John G. Proakis and Masoud Salehi. I’m studying communication systems and looking for some reference sources. Thanks a lot.

@Leo: Well, I do not think that Fundamentals of Communication Systems may not be available online. I can point you to another book

Signal Processing for Communications – http://www.sp4comm.org/webversion.html. Nice read.

Good luck in your studies

I am a member and I have still not received the promised pdf. Krishna can you please tell me how can I get it

@ communications engineer : Typically, the e-Book is offered for subscribers who have opted in to receive posts over email. Anyhow, since you have already joined Google FriendConnect, I have send you the link over email.

respected sit,

your site is very useful for me

thank you

premalatha

Θα ήθελα πάρα πολύ αυτό το βιβλίο. Ευχαριστώ

@Fred: Is this a question or a statement? 😉

Simulate 8-PSK (k = 3, M = 8) using the mapping below:

There is no need to generate individual bits. The input to the detector is r = s + n where all numbers are complex. The input, s, is a complex number with magnitude A and phase

determined by the random input. To generate complex noise, you need two independent

Gaussian random variables. Alternatively, you can model all signals as two-dimensional

vectors. The detector finds the closest signal point to r.

For this simulation, SNRb = (A^2)/6. Simulate the performance the system for SNRb = 7, 8,

9, 10, 11 dB and find the symbol and bit error rates. The printout format is:

System:8-PSK; SNRb=___(dB); #Bits transmitted=___ (i N/10, i = 1,…,10); SER=___; BER=___

Compare your results with the theoretical estimates of symbol and bit error rates. Plot the

theoretical estimates (curves) and the simulated error rates (points) in one figure.

@Hyde: I emailed you the download URI.

would u mind to send me the download URL,, thanks alot..

@Shreyas: I had sent you the download URI over email.

Hi Krishna,

I would really appreciate if you could send me the ebook.

Thanks,

Shreyas

@Steven: I mailed you the download URL. Happy learning.

please send me the link for free book on AWGN

@Ramu: I just send you instructions over email.

Hi Krishna,

I would really appreciate if you could send me the ebook too.

Thanks,

John.

@John: I just mailed you the download instructions. Plz check.

Please send me the ebook, thankS!

@jamal: I have sent you the download URL over email.

Atsiuskite ebook Matlab

Hello

I need matlab code for generating the diamond curve in figure 1 in the 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

@ShanmukhaPriya: Sorry, I do not have time to simulate that work. Good luck.

I would also like to download this book but I don’t now how to get link. Thanks in advance.

@Dule: I emailed you the download instructions

Hi Krishna,

How do I get a copy of the ebook, since I am already a member ?