How can I resample at original rate

Hello,

I transmit a signal over a channel that time-scale and time-shift the transmit signal. To generate the received signal I first resample the transmit signal as:

d_Res=resample(d,p1,q1);

and then apply time shifting to it, which results in the received signal v. I want to sample v at the original rate of d. How can I do that?

Thanks in advance

[S. David]

When I resample an additive white Gaussian noise (AWGN) process at different rates, all the resulting samples will be independent, right?

Answer 1

[Star Strider]

Not really certain (in spite of our earlier conversations) that I completely understand.

Would

vr = resample(v, q1,p1)

produce the result you want?

[S. David]

If we transmit a rectangular pulse g(t) of duration Ts, the time-scaled signal will be g(t*[1+a]) of duration Ts'=Ts/(1+a), where Ts=1/fs is the sample duration and fs is the sampling rate. So, the effective sampling rate of the received signal is fs'=(1+a)*fs.

So, if p and q are factors that resample the original signal (1+a) times the original sampling rate fs, then g(t*[1+a]) can be generated as:

gResCh=resample(g,p,q);

where g is the original signal. Now gRes has a sampling rate of fs', so to compensate for the scaling we would resample gResCh at rate fs'/(1+a) to recover g(t). In MATLAB this can be written as:

gResRec=resample(g,q,p);

I don't want to compensate for the scaling, so I think gResCh is what I want, because the received signal is g(t*[1+a]) and sampling at rate fs will give gResCh. Right?

I don't have a match yet, but I think I have a problem generating the equivalent channel. I just want to make sure that the noise-free received signal is correct given the above.

[S. David]

It appears to have a constant shift as in the figure attached.

[Star Strider]

‘When I resample an additive white Gaussian noise (AWGN) process at different rates, all the resulting samples will be independent, right?’

I would assume in that situation that the samples would be i.i.d., yes.

With respect to the constant shift, I thought that is what you want.

[S. David]

I didn't understand what you mean by " With respect to the constant shift, I thought that is what you want ", but what happens physically is the following:

1. The channel time-scale the signal
2. The channel time-shift the time-scaled signal.
3. AWGN is added at the front-end receiver.
4. The receiver resample the received signal.

So, if I resample the received signal as many times as I want, given that each time I do it at a different rate, then all the noise samples will remain independent and identically distributed (i.i.d), right?

I want to make sure because I am using minimum mean square error (MMSE) equalizer which requires the knowledge of the noise covariance matrix.