Add delay to signal

I would like to simulate on Matlab a PAM signal which, travelling on a time-dispersive communication channel, contains intersymbol interference (ISI).

I have 'built' the transmitter by generating:

• M-PSK signal/sequence
• designing the impulse response p(t) of a root of raised cosine
• upsampling
• filtering

So I have the signal ready to transmit ... but without ISI. Now I want to simulate/insert 'L' echoes of the time-dispersive channel.

In the transmission I don't want to have p(t) but a delayed version of it .. so I want the impulse p(t) to be born already with echoes .. for example like this: p(t)+a1*p(t-Ts) with Ts=symbol interval. Therefore I must linearly combine 2 or 3 appropriately delayed replicas of the impulse response in the absence of ISI. But how do I achieve it?

The professor advised me to use "the function that implements the filter impulse response with delay "... but what function is he talking about?

Any advice is helpful. Thanks!

[Paul]

What is the form of p(t)? Is it a vector of values corresponding to equally spaced points in time?

[kale]

Yes, on Matlab it's a vector of values corresponding to equally spaced points in time

But I honestly do not know what p(t) is in this process .. yes it could be the vector of coefficients of the raised root cosine but I'm not very sure.

In fact, seq_filtered = filter(coefficient, 1, seq_upsampled)

[Paul]

Is Ts an integer multiple of the spacing between the time points?

[kale]

No it's the symbol interval, for example 1 second

[Paul]

So p(t) is a continuous-time signal represented by samples p[n] = p(n*T) where T is the time spacing. Ts is an arbitrary delay, not necessarily an integer mutliple of T. How do you want to represent p(t) between samples of p[n]. Linear interpolation? Something else?

Reading through this comment thread, I'm not sure if where I'm headed with this is where you should be going ...

[kale]

p(t) is what is theoretically referred to as the 'pulse' for the PAM signal, i.e. I have chosen (as is often the case) the root raised cosign.

I didn't actually define it explicitly on Matlab, I wrote these steps:

M = 8;
psklev = exp(1j*2*pi*(0:M-1)/M); % M-PSK levels
l = randi([0 M-1], Nsym, 1);     % generate Nsym random M-ary symbols
[lg, ~] = togray(l, mb);         % Gray mapping
seq = psklev(lg+1);             

coeff = rcosdesign(a=0.35, L=30, Ns=10, 'sqrt') % it's discrete, in fact I use the 'stem' command to print it out)
seq_upsample = upsample(seq, Nc=Ns)
seq_filt = filter(coeff, 1, seq_upsample); % this signal is ready to be transmitted, it does not contain ISI

These are the salient steps, but I can send you the whole live script if you need it.

To the seq_filt signal I must add intersymbol interference ... and the professor recommended me to delay the impulse p(t), already creating it with echoes. For example like this: p(t)+a1*p(t-Ts) with Ts=symbol interval.

In the code I have, I do not understand what p(t) is and how I should proceed to achieve this.

Answer 1

[Paul]

Maybe this will help:

Let's assume we have a simple FIR filter that takes the average of the current and two previous samples of a signal. That filter would have a transfer function of

H(z) = (1 + z^-1 + z^-2)/3;

In Matlab, we represent this as:

b = [1 1 1]/3; a = 1;

Suppose we have some input sequence, u[n]. Then we know that

Y(z) = H(z)*U(z)

Here we just use a random sequence

rng(100);
u = randn(100,1);

The output of our filter, y[n], in response to u[n] is

y = filter(b,a,u);

Suppose we want to delay the output y by 3 samples. Well, that means that the z-transform of the delayed output is z^-3*Y(z), or

z^-3*H(z)*U(z). So can mutiply H(z) by z^-3 and then run u through that. z^-3 is represented by

z3 = [0 0 0 1]; % 0 + 0*z^-1 + 0*z^-2 + 1*z^-3;

Polynomial multiplication is implemented with conv. So the delayed signal is

y3 = filter(conv(z3,b),1,u);

Plot the output and delayed output, and their sum

subplot(211);plot([y y3])
subplot(212);plot(y+y3)

[kale]

Thank you for the example! Everything is now clearer.

So basically I have two filters:

- the first filter generates the coefficients (without delay) of RRC impulse given by coeff=rcosdesign(...)

- the second filter adds the delay/ISI to the coefficients generated by the first. So the output of the first filter is the input of the second one

Following your example, I focus only on the second filter (I have already implemented the first one) and my question now is:

if with the second filter I want to delay by 8 samples each of the samples/coefficients of the first filter RRC ..is it correct to do as I did here?

coeff = rcosdesign(alpha=0.35, L=30, Ns=10, 'sqrt') % Ideal! Without delay
% now I delay the the coeff by 3 samples:
z = [0 0 0 0 0 0 0 0 1]; % 0 + 0*z^-1 + 0*z^-2 + 0*z^-3 ... 1*z^-8;
coeff_delay = filter(z3,1,coeff);

subplot(2,1,1);
stem((-L/2:1/Nc:L/2),coeff)
title('RCC Impulse response :)');
subplot(2,1,2);
stem((-L/2:1/Nc:L/2),coeff_delay);
title('RCC Impulse response + delay :(');

Do you think I should add these two signals together in the rest of the programme? (as you did in y+y3). Or should I just use the second one? (i.e. y3)

Thanks!

[Paul]

Maybe it's just a terminolgy thing, but I don't understand what you mean by delaying coefficients of a filter. Signals get delayed, not filter coefficients.

Anyway, it sounds like you have a signal u[n] going through a filter H(z) = b(z)/a(z)

y = filter(b,a,u);

Now you want to take that output, and delay it by 8 samples.

b8 = [0 0 0 0 0 0 0 0 1]; % 0 + 0*z^-1 + 0*z^-2 + 0*z^-3 ... 1*z^-8;
y8 = filter(b8,1,y);

It sounds (pun intended) like you want to add y and y8 together to get some sort of echo effect on the original signal. But I'm not really sure what you're trying to do.