DSP LOG - Page 9 of 12 - Signal Processing

Article in DSPDesignLine.com: M-QAM symbol error

Its been a nice week for me, wherein I guest posted an article in DSPDesignLine.com. 🙂

The article derives the theoretical symbol error rate for M-QAM modulation. The theoretical results are further supplemented by Matlab/Octave simulation scripts.

Those who are familiar with derivation of symbol error rate for 16-QAM modulation will find the equations easy to interpret. As we did for 16-QAM,

Continue reading “Article in DSPDesignLine.com: M-QAM symbol error”

Bit error rate for 16PSK modulation using Gray mapping

In this post, let us derive the theoretical bit error probability for 16PSK modulation using Gray coded mapping. For deriving the equation, we will refer material from the following posts:

(a) Symbol Error Rate for 16PSK

(b) Gray code to Binary code conversion for PSK

As discussed in the previous posts, the key feature of Gray code is that adjacent symbols differ by only one bit. The 16PSK constellation with Gray mapping can be as shown in the figure below.

Continue reading “Bit error rate for 16PSK modulation using Gray mapping”

Gray code to Binary conversion for PSK and PAM

Given that we have discussed Binary to Gray code conversion, let us discuss the Gray to BInary conversion.

Conversion from Gray code to natural Binary

Let $g[n-1:0]$ be the equivalent Gray code for an $n$ bit binary number $b[n-1:0]$ with $j$ representing the index of the bit.

1. For $j=n-1$ ,

$b[n-1] = g[n-1]$ i.e, the most significant bit (MSB) of the Gray code is same as the MSB of original binary number.

2. For $j=n-2\mbox{ to }0$ ,

$b[j]=b[j+1]\oplus g[j]$ i.e, $j^{th}$ bit of the Binary number is the exclusive-OR (XOR) of ${j}^{th}$ of the bit of the Gray code and ${j+1}^{th}$ of the bit of the binary number. Continue reading “Gray code to Binary conversion for PSK and PAM”

Binary to Gray code conversion for PSK and PAM

In this post, let us try to understand Gray codes and their usage in digital communication. Quoting from Wiki entry on Gray code [Gray-Wiki],

The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one digit.

In a digital communication system, if the constellation symbols are Gray encoded, then the bit pattern representing the adjacent constellation symbols differ by only one bit. We will show in another post that having this encoding structure gives a lesser probability of error than the ‘natural binary ordering’. However, in this post, let us try to figure out the conversion of natural binary representation to Gray code. Continue reading “Binary to Gray code conversion for PSK and PAM”

OT: Migration to a Deep Blue template

After almost 6 months with the Smashing Theme template, its time for a change. Recently I stumbled upon the Deep Blue template from DailyBlogTips. I liked the clean blue-green-white combination and and felt it might be a good fit for www.dsplog.com. Hope you agree.

One key feature which I like is addition of tag cloud in the sidebar. Hopefully navigating to the relevant posts became simpler. Have a look around.

Of course, your valuable comments and criticism are welcome. 🙂

Eye diagram with raised cosine filtering

We have discussed about probable transmit pulse shaping filter and have observed that raised cosine filtering filtering allows a simpler implementation, albeit at the cost of increased bandwidth. Let us know understand the eye diagram, which is a useful graphical tool to quantify the degradation of the signal due to filtering.

Eye diagram

An eye diagram is generated in an oscilloscope operating in the persistence mode by observing the output of the filter with the symbol timing serving as the trigger. The observation window can be set as 2 times the symbol period. (Refer. Section 5.1.3 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

Continue reading “Eye diagram with raised cosine filtering”

MCDES 2008 at Indian Institute of Science, Bangalore

Thanks to Prof. K V S Hari, I was informed that the centenary year for Indian Institute of Science (IISc), Bangalore starts on May 27th 2008.

As part of the event, a conference with the theme Managing Complexity in a Distributed World (MCDES) is to be held at IISc from May27th to 31st 2008. Continue reading “MCDES 2008 at Indian Institute of Science, Bangalore”

Raised cosine filter for transmit pulse shaping

In the previous post on transmit filtering using Nyquist pulse, we had briefly learned that the information symbol $a_m$ with a symbol period $T$ can be transmitted without inter symbol interference (ISI) by using Nyquist pulse,

$g(t) = \frac{sin(\pi t/T)}{\pi t/T},\mbox{ } t=-\infty \mbox{ to } +\infty$ .

The resultant waveform is ideally bandlimited to frequencies from $-\frac{1}{2T}$ Hz to $+\frac{1}{2T}$ Hz.

However, in typical transmission schemes, we do not hear of pulse shaping using sinc() filters. Rather, pulse shaping using raised cosine filter is frequently used. In this post, objective is to understand the motivation behind using raised cosine filtering for pulse shaping.

Continue reading “Raised cosine filter for transmit pulse shaping”

OT: Prof. Randy Pausch’s lecture in Oprah show

In the past, I have wondered about discussing personal thoughts in this blog. The answer in my mind was NO and I ve been focusing only on technical topics till date. However, there is a change of mind, thanks to my friend Manoj.

Thanks to him, I happened to see Prof. Randy Pausch’s brief 10 minute lecture in Oprah show. The contents of video is very powerful and motivated me to have a rethink. Please find the YouTube video embedded below.

Continue reading “OT: Prof. Randy Pausch’s lecture in Oprah show”

Transmit pulse shaping filter – rectangular and sinc (Nyquist)

In the previous post on I-Q modulator and de-modulator, we had briefly mentioned that the a baseband PAM transmission can be modelled as

$s(t) = \sum_{m=-\infty}^{\infty}a_mg(t-mT)$ , where

$T$ is the symbol period,

$a_m$ is the symbol to transmit,

$g(t)$ is the transmit filter,

$m$ is the symbol index and

$s(t)$ is the output waveform.

In this post, the objective is to understand the properties of the transmit filter $g(t)$ i.e. to find out a filter which occupies the minimum required bandwidth while ensuring inter-symbol-interference (ISI) free transmission of the information symbol $a_m$ .

Continue reading “Transmit pulse shaping filter – rectangular and sinc (Nyquist)”

IQ modulation and demodulation

Post describes about the need for I-Q modulation by comparing the spectral efficiency of passband PAM and passband QAM.

PAM (Pulse Amplitude Modulation) is one of the simplest ways to send digital data. In this post, we start with baseband PAM, look at its spectral efficiency, and then move on to passband PAM — which turns out to be less efficient. To fix that, we explore how QAM (Quadrature Amplitude Modulation) improves efficiency by using both sine and cosine carriers. This leads to I/Q modulation, and the corresponding IQ demodulation with Python code.

The content is based onSec 5.2.1 of [DIG-COMM-BARRY-LEE-MESSERSCHMITT].

Table of Contents

Baseband PAM transmission

Consider a simple baseband transmission where the information is sent by modulating a pulse. This can be represented as

$s(t) = \sum_{m=-\infty}^{\infty}a_mg(t-mT)$ , where

$T$ is the symbol period,
$a_m$ is the symbol to transmit,
$g(t)$ is the transmit pulse shaping filter,
$m$ is the symbol index and
$s(t)$ is the output waveform.

Pictorially the same can be represented as,

Figure: Baseband PAM transmission

Minimum bandwidth for ISI free transmission

According to the Nyquist zero ISI criterion, to transmit data without intersymbol interference (ISI), the pulse shaping filter $g(t)$ must satisfy,

$g(nT) = \begin{cases} 1, & n = 0 \\ 0, & \text{otherwise} \end{cases}$

This ensures that each symbol appears cleanly at its own sampling instant without interference from neighboring symbols. For this condition to hold and still remain bandlimited to $\frac{1}{2T}$ , the ideal pulse shape is the sinc function:

$\begin{array}{lll} g(t) &=& \dfrac{\sin\left(\pi t / T\right)}{\pi t / T}, \quad t \in (-\infty, +\infty) \\ &=& \mathrm{sinc}\left(\dfrac{t}{T}\right) \end{array}$

This result comes from the fact that, with ideal sinc pulse shaping, the entire symbol stream can be perfectly reconstructed if the channel passes all frequencies up to $\frac{1}{2T}$ . This is also consistent with the Nyquist–Shannon sampling theorem, which states that to fully reconstruct a signal sampled at $f_s$ , the signal must be bandlimited to $\frac{f_s}{2}$ .

Thus, if we use ideal (bandlimited) pulses like sinc functions, the entire baseband PAM signal can be confined within:

$\text{Bandwidth} = \frac{1}{2T} \text{Hz}$

Figure: Spectrum of baseband PAM (with symbol rate $\frac{1}{T}$ )

Spectral Efficiency

Spectral efficiency is a measure of how efficiently a communication system transmits data over a given bandwidth. It is defined as the number of bits transmitted per second per unit bandwidth, typically expressed in bits/second/Hertz.

$\text{Spectral Efficiency} = \frac{\text{Bit Rate (bits/sec)}}{\text{Bandwidth (Hz)}}$

Assuming that PAM signal $a_m$ uses $N$ distinct amplitude levels, each symbol carries $\log_2(N)$ bits. When each symbol takes $T$ seconds to transmit, the bit rate of the system is:

$\text{Bit Rate} = \frac{\log_2 N}{T} \quad \text{bits/second}$

Substituting the bit rate and bandwidth into the formula for spectral efficiency, we get the spectral efficiency of baseband PAM as.

$\Gamma_{\text{BB-PAM}} = \frac{\log_2 N}{T} \cdot \frac{1}{$\frac{1}{2T}$} = 2 \log_2 N \quad \text{bits/second/Hertz}$

Code

Python code for passband PAM transmission with pulse shaping by rectangular filter and sinc filter.

Passband PAM transmission

Now, consider that the baseband PAM signal is upconverted to a carrier frequency $f_c$ by multiplication by a cosine carrier. The transmitted passband signal is:

$\hat{s}(t)=\sqrt{2}\cos$2\pi f_c t$\sum_{m=-\infty}^{\infty}a_m g\(t-mT)$

The spectrum of the passband PAM is as shown below,

Figure: Spectrum of passband PAM

When the baseband signal is multiplied by $f_c$ , this modulation shifts the entire spectrum up to be centered around the carrier frequency $f_c$ .

The frequency range of the signal becomes:

$f_c - \frac{1}{2T} \) \text{ to } \( f_c + \frac{1}{2T}$

Thus, the total bandwidth needed by passband PAM is,

$\text{Passband Bandwidth} = \left(f_c + \frac{1}{2T}\right) - \left(f_c - \frac{1}{2T}\right) = \frac{1}{T}$

Spectral Efficiency of passband PAM

As earlier, assuming that PAM signal $a_m$ uses $N$ distinct amplitude levels, each symbol carries $\log_2(N)$ bits. When each symbol takes $T$ seconds to transmit, the bit rate of the system is:

$\text{Bit Rate} = \frac{\log_2 N}{T} \quad \text{bits/second}$

Substituting the bit rate and bandwidth into the formula for spectral efficiency, we get the spectral efficiency of passband PAM as,

$\Gamma_{\text{PB-PAM}} = \frac{\log_2 N}{T} \cdot \frac{1}{$\frac{1}{T}$} = \log_2 N \quad \text{bits/second/Hertz}$

Can see that spectral efficiency of passband PAM is half of baseband PAM.

IQ Modulation (QAM for improving spectral efficiency)

As seen earlier, passband PAM suffers from reduced spectral efficiency. To improve spectral efficiency, we exploit the orthogonality of sine and cosine carriers. This leads to Quadrature Amplitude Modulation (QAM), where two independent data streams are modulated:

to a cosine wave (in-phase component, I) $a^I_m$
to a sine wave (quadrature component, Q) $a^Q_m$

Because sine and cosine are orthogonal over a symbol period $\int_T\sin$\omega t$\cdot \cos$\omega t$= 0$ , these two streams do not interfere and can be perfectly separated at the receiver. As a result, QAM doubles the data rate over the same bandwidth compared to passband PAM.

The transmit signal with passband QAM is represented as,

$\bar{s}(t) = \sqrt{2} \cos(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^I_m g(t - mT) - \sqrt{2} \sin(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^Q_m g(t - mT)$

where,

$a^I_m$ and $a^Q_m$ are the in-phase and quadrature data symbols, respectively
$g(t)$ is the transmit pulse shaping filter, and
$f_c$ is the carrier frequency

This signal requires the same bandwidth as passband PAM, however carries twice the information.

Note :

The factor of $\sqrt2$ is included to normalize the total power of the passband QAM signal to unity. As cosine and sine carriers each contribute only half the power $\frac{1}{2}$ when modulating a baseband signal, multiplying each arm by $\sqrt2$ ensures that the modulated in-phase and quadrature components retain their original baseband power.

For notational convenience, we express the passband signal using its complex baseband equivalent,

$\bar{s}(t)=\sqrt{2}\Re \left[ \tilde{s}(t)e^{j2\pi f_c t} \right]$ ,

where

$\tilde{s}(t)=\sum_m \tilde{a}_m g(t - mT)$
$\tilde{a}_m=a^I_m+j\cdot a^Q_m$

This forms the I-Q modulator circuit.

Figure: IQ modulator

IQ demodulation

The received passband signal is split into two branches and multiplied with cosine (in-phase, I) and sine (quadrature, Q) carriers. Each branch is then passed through a matched low-pass filter, typically matched to the transmit pulse shape, to extract the baseband I and Q components. The resulting streams are then independently sampled and demodulated to recover the transmitted symbols.

Downconversion

Let the received passband signal

$\begin{array}{lll}r(t) & = & \bar{s}(t)\\ &=&\sqrt{2} \cos(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^I_m g(t - mT) - \sqrt{2} \sin(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^Q_m g(t - mT)\\ &=&\sqrt{2} \cos(2\pi f_c t) I(t) - \sqrt{2} \sin(2\pi f_c t) Q(t) \end{array}$

where,

$I(t) = \sum_{m=-\infty}^{\infty} a^I_m g(t - mT)$

$Q(t) = \sum_{m=-\infty}^{\infty} a^Q_m g(t - mT)$

To extract the I and Q signals, multiply $r(t)$ with local carrier signals :

In-phase branch:

$\begin{array}{lll} I_{\text{branch}}(t) & = & r(t) \cdot \sqrt{2} \cos(2\pi f_c t) \\ & = & 2I(t)\cos^2(2\pi f_c t) - 2Q(t)\sin(2\pi f_c t)\cos(2\pi f_c t) \\ & = & 2I(t) \left( \frac{1}{2} + \frac{1}{2} \cos(4\pi f_c t) \right) - 2Q(t) \cdot \frac{1}{2} \sin(4\pi f_c t) \\ & = & I(t) + I(t)\cos(4\pi f_c t) - Q(t)\sin(4\pi f_c t) \end{array}$

Similarly, for the quadrature branch:

$\begin{array}{lll} Q_{\text{branch}}(t) & = & r(t) \cdot \sqrt{2} \sin(2\pi f_c t) \\ & = & 2I(t)\cos(2\pi f_c t)\sin(2\pi f_c t) - 2Q(t)\sin^2(2\pi f_c t) \\ & = & 2I(t) \cdot \frac{1}{2} \sin(4\pi f_c t) - 2Q(t) \left( \frac{1}{2} - \frac{1}{2} \cos(4\pi f_c t) \right) \\ & = & I(t)\sin(4\pi f_c t) - Q(t) + Q(t)\cos(4\pi f_c t) \end{array}$

After downconversion, each branch (I and Q) contains the desired baseband signal components, along with unwanted high-frequency terms centered around $2f_c$ .

Low-pass Filtering

To isolate the desired baseband signals, we apply a low-pass filter (LPF) to both the I and Q branches. This filter passes frequencies within the baseband bandwidth (usually $[{-1}/{(2T)}\text{ to } {+1}/{(2T)}]$ where is the symbol period .

Matched Filtering

After low-pass filtering, we apply a matched filter to each of the I and Q branches. The matched filter is designed to maximize the signal-to-noise ratio (SNR) at the symbol sampling instants. It is typically a time-reversed and conjugated version of the transmit pulse shape $g(t)$ .

If the transmit pulse was a rectangular or truncated sinc or raised-cosine shape, the matched filter is also a rectangular or truncated sinc or raised-cosine, respectively. This ensures:

Maximum energy concentration at symbol instants
Minimization or elimination of inter-symbol interference (ISI)

The output of the matched filter is then sampled every seconds (the symbol period) to extract the transmitted symbol values $a^I_m$ and $a^Q_m$ .

This sequence completes the demodulation chain for QAM and ensures that both spectral shaping and ISI suppression are achieved in the receiver.

The IQ demodulator can be visualized as shown in the figure below

Figure: IQ demodulator

Code

Python code for passband qam with rectangular and sinc pulse shaping filter, followed by low pass filtering and matched filtering.

When compared to the transmit symbols, can see that there is a spread in the received symbols. Using raised cosine or root raised cosine filters is a potential approache to mitigate this.

Summary

The post covers the following key aspects

Spectrum of Baseband PAM with a code example
Reduced Spectral Efficiency of Passband PAM
Sending information on orthogonal sine and cosine carriers to improve efficiency (passband QAM)
IQ modulator and corresponding demodulator for Passband QAM with code

Another approach for improving the spectral efficiency of passband PAM is to filter away half the bandwidth (which is not carrying any ‘extra’ information). This is called Single Sideband Modulation (SSB). Anyhow, as I-Q modulation is simpler to implement than circuit for filtering away half the spectrum, I-Q modulation stays. 🙂

Have any questions or feedback? Feel free to drop your feedback in the comments section. 🙂

Reference

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

Comparing 16PSK vs 16QAM for symbol error rate

In two previous posts, we have derived theoretical symbol error rate for 16-QAM and 16-PSK modulation schemes. The links are:

(a) Symbol error rate for 16-PSK

(b) Symbol error rate for 16-QAM

Given that we are transmitting the same number of constellation points in both 16-PSK and 16-QAM, let us try to understand the better modulation scheme among the two, i.e. to answer the following question:

For the same signal to noise ratio $\frac{E_s}{N_0}$ , will 16-PSK or 16-QAM give a lower symbol error rate?

Continue reading “Comparing 16PSK vs 16QAM for symbol error rate”

Blog on DSP

I happened to visit ‘The Digital Signal Processing Blog’, maintained by Mr. Andres Kwasinski, Ph. D.

In the blog one can find details about the upcoming IEEE conferences pertaining to communication and multimedia processing.
Further, in some of the posts, author shares his thoughts on topics like fixed point arithmetic (here) and wavelets (here) etc.

Thanks,
Krishna

Symbol Error Rate for 16PSK

In this post, let us try to derive the symbol error rate for 16-PSK (16-Phase Shift Keying) modulation.

Consider a general M-PSK modulation, where the alphabets,

$alpha_{16PSK} =\sqrt{E_s}\left\{1,\ e^{\frac{j2\pi}{M}},\ e^{\frac{j4\pi}{M}},\ \ldots,\ e^{\frac{j2\pi(M-1)}{M}} \right}$ are used.

(Refer example 5-38 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT])

Figure: 16-PSK constellation plot

Continue reading “Symbol Error Rate for 16PSK”