Tag: square-root raised-cosine pulse shaping

Key focus: Let’s learn how to simulate matched filter receiver with square root raised cosine (SRRC) filter, for a pulse amplitude modulation (PAM) system.

Simulation Model

A basic pulse amplitude modulation (PAM) system as DSP implementation, is shown in Figure 1 by adding an upsampler (), pulse shaping function () at the transmitter and a matched filter (), downsampler () combination at the receiver.

In this model, a random stream of source bits is first segmented into -bit wide symbols that can take any value from the set . The simulation code directly starts by generating a random set of symbols, that goes into the modulation mapper. Pulse amplitude modulation (MPAM) mapping and de-mapping, described in sections 5.3.1 and 5.4.1, are considered here for simulation. An MPAM modulator maps the -bit information symbols to one of the distinct signaling levels. The MPAM modulated symbols are shown in Figure 2.

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

%Program: MPAM modulation
N = 10ˆ5; %Number of symbols to transmit
MOD_TYPE = 'PAM'; %modulation type
M = 4; %modulation level for the chosen modulation MOD_TYPE
d = ceil(M.*rand(1,N)); %random numbers from 1 to M for input to PAM
u = modulate(MOD_TYPE,M,d);%MPAM modulation
figure; stem(real(u)); %plot modulated symbols

Each MPAM modulated symbol should last for some duration called symbol time, denoted as . Each modulated symbol will go through a discrete time pulse shaping filter whose impulse response is spaced sample, where denotes the sampling period. To do this, the incoming symbols from the modulation mapper need to be converted to discrete time impulse train by upsampling them by a factor (as per the upsampling equation given here ). The upsampler inserts zeros between each modulated symbols. In practice, is chosen as integral multiples of 4. The upsampler/oversampled output is shown in Figure 3.

%Program: Upsampling
L=4; %Oversampling factor (L samples per symbol period)
v=[u;zeros(L-1,length(u))];%insert L-1 zero between each symbols
%Convert to a single stream
v=v(:).';%now the output is at sampling rate
stem(real(v)); title('Oversampled symbols v(n)');

In order to fill-in proper values in place of the inserted zeros, interpolation is performed by a pulse shaping filter by convolving the output of the upsampler and the pulse shaping function. The pulse shaping function needs to satisfy Nyquist criterion for zero ISI, otherwise, aliasing effect will wreak havoc. If the amplitude response of the channel is flat and if the noise is white, then the amplitude response of the pulse shaping function can be split equally between the transmitter and receiver. For this simulation the desired Nyquist pulse shape is a raised-cosine pulse shape and the task of raised-cosine filtering is equally split between the transmit and receive filters. This gives rise to square-root raised-cosine (SRRC) filters at the transmitter and receiver. This is a matched filter system, where the receive filter is matched with the transmit pulse shaping filter.

A matched filtering system is a theoretical framework and it is not a specific type of filter. It offers improved noise cancellation by improving the signal noise ratio at the output of the receive filter. The implementation starts with the design of an SRRC filter with roll-off factor . The SRRC filter length is influenced by the parameter – the span of the filter length in units of symbols and the oversampling factor .

Filters will not produce instantaneous output and they take sometime to produce the output. That is, the output of the filter is shifted in time with respect to the input. For symmetric FIR filters of length , the filter delay is . Apart from returning the SRRC pulse function, the filter design function given in this section returns the filter delay. Filter delays are useful in determining the appropriate sampling instances at the receiver. The modulated symbols at the transmitter are passed through the designed filter and the response of the filter is plotted in Figure 3 (right).

%Program: SRRC pulse shaping
%----Pulse shaping-----
beta = 0.3;% roll-off factor for Tx SRRC filter
Nsym=8;%SRRC filter span in symbol durations
L=4; %Oversampling factor (L samples per symbol period)
[p,t,filtDelay] = srrcFunction(beta,L,Nsym);%design filter
s=conv(v,p,'full');%Convolve modulated syms with p[n] filter
figure; plot(real(s),'r'); title('Pulse shaped symbols s(n)');

The pulse shaped signal samples are sent through an AWGN channel, where the transmitted samples are added with noise samples that are generated according to the required (refer AWGN noise model given in this post). The received signal that is corrupted with AWGN noise is shown in Figure 4 (left).

%Program: Adding AWGN noise for given SNR value
EbN0dB = 10; %EbN0 in dB for AWGN channel
snr = 10*log10(log2(M))+EbN0dB; %Converting given Eb/N0 dB to SNR
%log2(M) gives the number of bits in each modulated symbol
r = add_awgn_noise(s,snr,L); %AWGN , add noise for given SNR, r=s+w
%L is the oversampling factor used in simulation
figure; plot(real(r),'r');title('Received signal r(n)');

For the receiver system, we assume that the ADC in the receiver produces an integer number of samples per symbol (i.e, is an integer). In practice, this is not always the case and thus a resampling filter is often included in real world designs. In the discrete time model, the received samples are passed through a matched filter, whose impulse response is matched to the impulse response of the pulse shaping filter as . Since the SRRC pulse is symmetric, we will be using the same SRRC pulse shaping function for the matched filter. The received samples are convolved with the matched filter and the output of the matched filter is shown in Figure 4 (right).

Refer the book Wireless Communication Systems in Matlab for the program on how to perform matched filtering

Next, we assume that the receiver has perfect knowledge of symbol timing instants and therefore, we will not be implementing a symbol timing synchronization subsystem in the receiver. At the receiver, the matched filter symbols are first passed through a downsampler that samples the filter output at correct timing instances.

The sampling instances are influenced by the delay of the FIR filters (SRRC filters in Tx and Rx). For symmetric FIR filters of length , the filter delay is . Since the communication link contains two filters, the total filter delay is . Therefore, the first valid sample occurs at position in the matched filter’s output vector ( is added due to the fact that Matlab array indices starts from 1). The downsampler that follows, starts to sample the signal from this position and returns every symbol. The downsampled output, shown in Figure 5, is then passed through a demodulator that decides on the symbols using an optimum detection technique and remaps them back to the intended message symbols.

%Program: Symbol rate sampler and demodulation
%------Symbol rate Sampler-----
uCap = vCap(2*filtDelay+1:L:end-(2*filtDelay))/L;
%downsample by L from 2*filtdelay+1 position result by normalized L,
%as the matched filter result is scaled by L
figure; stem(real(uCap)); hold on;
title('After symbol rate sampler $\hat{u}$(n)',...
'Interpreter','Latex');
dCap = demodulate(MOD_TYPE,M,uCap); %demodulation

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Topics in this chapter

Pulse Shaping, Matched Filtering and Partial Response Signaling ● Introduction ● Nyquist Criterion for zero ISI ● Discrete-time model for a system with pulse shaping and matched filtering  □ Rectangular pulse shaping  □ Sinc pulse shaping  □ Raised-cosine pulse shaping  □ Square-root raised-cosine pulse shaping ● Eye Diagram ● Implementing a Matched Filter system with SRRC filtering  □ Plotting the eye diagram  □ Performance simulation ● Partial Response Signaling Models  □ Impulse response and frequency response of PR signaling schemes ● Precoding  □ Implementing a modulo-M precoder  □ Simulation and results