# Waveform obtained due to random BPSK data

Hi there,

I am trying to obtain the waveform due to random binary phase-shift keying. the graph is similar to a sinc/sampling function. Can someone help how to go about it?

regards
kautilya

berkeman
Mentor
Hi there,

I am trying to obtain the waveform due to random binary phase-shift keying. the graph is similar to a sinc/sampling function. Can someone help how to go about it?

regards
kautilya

The "waveform" or the spectra? Can you please put your question in context, and give more details about what you are looking for and why?

jbunniii
Homework Helper
Gold Member
Hi there,

I am trying to obtain the waveform due to random binary phase-shift keying. the graph is similar to a sinc/sampling function. Can someone help how to go about it?

regards
kautilya

I assume you mean the power spectral density, since you said it looks similar to a sinc function. (Actually sinc squared.)

In what way do you want to "obtain" it? Do you want to derive an equation for it, or produce a plot of it using a simulation, or measure it on a spectrum analyzer, or what?

The equation is easy to derive, if you recognize that the autocorrelation function corresponding to a random BPSK signal (at baseband) is a triangle, the base of which is twice the bit length, and that the power spectral density is the Fourier transform of the autocorrelation function.

Hi there

Thanks for the reply. When i said i wanted to obtain it, yes, i really meant deriving the equation of the power spectral density function and also plot the function using MATLAB simulation from the random data of BPSK.

Can you help me regarding this?

regards
kautilya

jbunniii
Homework Helper
Gold Member
Hi there

Thanks for the reply. When i said i wanted to obtain it, yes, i really meant deriving the equation of the power spectral density function and also plot the function using MATLAB simulation from the random data of BPSK.

Can you help me regarding this?

regards
kautilya

To derive it, like I said, start with the autocorrelation function, which is

R(tau) = E[x(t) x(t + tau)]

Without too much difficulty, you should be able to show that this is a triangular function between -T and T (where T is the bit length), centered at 0, and zero outside that interval. This assumes that a +1 bit and -1 bit are equally likely and that the bit values are statistically independent from one another. (Actually uncorrelated would suffice.)

If you're stuck on that part, please show us what equation you are assuming for x(t), and how far you are able to get.

This all presumes that x(t) is wide-sense stationary; otherwise the autocorrelation function will depend on two parameters, not just one. Wide-sense stationarity should be true in your case unless you're making an unusual assumption of some kind, but you should verify that this is true.

Once you have that R(tau) is a triangle function, just take its Fourier transform to obtain a sinc^2. Either you know this reflexively based on having it hammered into you in a signals and systems course, or you can derive it by expressing R(tau) as the convolution of a simpler function with itself, and remembering what convolution in the time domain corresponds to in the frequency domain.

Finally, for simulating all of this in Matlab, I assume you can generate a vector containing a simulated BPSK time domain signal. (Either with or without noise added.) I'll assume you have B samples per bit and M bits, so the vector contains BM samples. You'll then want to repeat this process N times (obtain M random BPSK vectors) so you can do a Monte Carlo approximation to the expected value in the defining equation for R(tau).

You can then estimate the power spectral density in one of two ways.

(1) Compute x(t)x(t-tau) for each BPSK vector and for as many values of tau as desired (tau is the offset between samples in the product, and has the same granularity as your sampling rate). For each tau, you can accumulate across all the t values because of the wide-sense stationary assumption. For each BPSK vector this will produce a sample autocorrelation function, call it R_n(tau), where n varies from 1 to N. Then simply average the R_n(tau)'s from 1 to N for each tau, resulting in R(tau). You can then take the Fourier transform using Matlab.

(2) You can directly compute the Fourier transform of each BPSK vector, and then take its magnitude squared. Repeat this process for each BPSK vector, and then average the magnitude-squared FFT's. This should result in the same power spectral density, except probably for a scale factor.

Hi there.

Thanks a lot for the reply jbunnii.

I understood your point regarding this. can u show me how do i generate this equation using a random data / vector without simulation? i mean mathematically can u derive the equation for me using random data bits?

i will highly appreciate it.

regards
kautilya

berkeman
Mentor
Hi there.

Thanks a lot for the reply jbunnii.

I understood your point regarding this. can u show me how do i generate this equation using a random data / vector without simulation? i mean mathematically can u derive the equation for me using random data bits?

i will highly appreciate it.

regards
kautilya

If this is for homework or coursework, then no, he should not derive the equation for you. That would be against the PF Rules.

What is the context of your question? What is the application? Is this for school work?

Hi there.

No, this is not a homework or coursework. This is a study-oriented university project i am carrying out on digital modulation techniques. I am getting stuck on this derivation part where we assume random data from BPSK. Can u pls help me derive the equation and hence plot the power spectral density for this case?

will appreciate it.

regards
kautilya

jbunniii
Homework Helper
Gold Member
Hi there.

No, this is not a homework or coursework. This is a study-oriented university project i am carrying out on digital modulation techniques. I am getting stuck on this derivation part where we assume random data from BPSK. Can u pls help me derive the equation and hence plot the power spectral density for this case?

will appreciate it.

regards
kautilya

Why don't you show us what you have done so far? Also, a derivation of this exact result should be in just about any textbook on digital communication systems. You can also find it in various engineering-oriented books on probability and random processes or on statistical signal processing/estimation theory.

Hi there.

I appreciate your reply. but the fact is i havent started or done anything regarding this. i was just reading material on this but my concepts arent clear still. with ur help i might be able to go a long way in this project. pls start off for me. i ll get an idea and can carry on further based on my understanding. pls start off with some random BPSK data.

will highly apprecite it. i have to submit it in 2-3 days. so it is urgent.

regards

kautilya

berkeman
Mentor
Hi there.

I appreciate your reply. but the fact is i havent started or done anything regarding this. i was just reading material on this but my concepts arent clear still. with ur help i might be able to go a long way in this project. pls start off for me. i ll get an idea and can carry on further based on my understanding. pls start off with some random BPSK data.

will highly apprecite it. i have to submit it in 2-3 days. so it is urgent.

regards

kautilya

We do not do your university work for you. You have been given quite good help in this thread so far. Re-read it, and look in the communication books as suggested above. Get used to doing your own work. You say it's not homework, but it's due in two days. Right.