# BPSK modulation under bandlimited channel

I modulated signal using BPSK modulation and I have series of -1 and 1. Now I have to pass these signal from band limited channel that has frequency response of the form $$\frac{s^2}{as^5+bs^4+cs^3+ds^2+es+f}.$$ Now I am thinking to convolve my -1 and 1 with time domain version of the above frequency response. I defined $a$, $b$, $c$... as a symbolic variable and I have their numerical value however I need to have my impulse response in terms $a$, $b$, $c$... Therefore I would like to know:

1. How to find impulse response when the frequency response is order 5 in denominator? Matlab seems to give me error message when I leave the expression in terms of symbolic coefficients.

2. Is my reasoning correct to multiply square pulse shape with a carrier and then convolve it with the impulse response of the channel?

3. Also what do you think is a good way to demodulate the signal? would that be sufficient to multiply the received signal by my basis vector like cosine or sine?

• Can you clarify whether the channel frequency response is a function of $W$ or a function of $f$?
– MBaz
Jun 25, 2015 at 14:10
• it is based on w. f is constant like a and b. Jun 26, 2015 at 3:53
• Your expression for the frequency response does not make much sense, at least not if $W$ is interpreted as a frequency or angular frequency. If $W$ was a frequency variable, then your channel would be non-causal and complex valued, i.e. for a real-valued input signal it would generate two output signals (real part and imaginary part), and the channel output would depend on past as well as on future inputs. Maybe $W$ is a complex Laplace transform variable, like what most people would call $s$ or $p$? Please clarify, otherwise your question cannot be answered. Jun 26, 2015 at 7:09
• @MattL. You are right that is my transfer function and should be based on s. so my w should be really i*w. Jun 26, 2015 at 21:16
• There is no analytic expression for the channel impulse response in terms of $a,b,\dots$, simply because you would need an analytic expression for the roots of a 5th order polynomial (the poles of the transfer function), which does not exist. What you'd need to do is compute (numerically) a partial fraction expansion given the numerical values of $a,b,\ldots$. From there you get the impulse response as a sum of weighted exponentials. Jun 27, 2015 at 6:50

1. Matlab can't handle symbolic calculations by itself. You need to use a symbolics-capable package, such as Matlab's own symbolic toolbox, Sage, SymPy, Maple, Mathematica, etc.

2. No, it is not correct. The sequence of -1 and 1 is a sequence of symbols, but what is actually transmitted over the channel is an analog signal. Between the symbols and the channel you need a pulse-shaping filter, which determines the transmitted signal's bandwidth and spectral characteristics.

3. The receive process will depend on the channel's frequency response. Say your transmitted signal has bandwidth $B$. If the channel response is (mostly) flat over that bandwidth, then the channel doesn't distort the signal and you can recover the information present in the signal by matched filtering. On the other hand, if the channel's response is not flat, then the signal will be distorted. In this case, you can perform channel equalization followed by matched filtering. A channel equalizer is a filter that is the "inverse" of the channel; the equalizer's output is an undistorted signal.

I've used a few concepts without explaining them, like pulse shaping and matched filtering. There are already many answers in this site regarding those concepts. If you need further help with these, please leave a comment.

• can you please explain a little more about modulation technique I have 1 and -1 I created a square pulse then I multiplied it by sin(2*pift) as a carrier then should I convolve it with impulse of my channel ? Jun 26, 2015 at 3:55
• (Note that, in your question, you don't mention multiplying the {-1,1} sequence by a sine carrier.) What you are doing is BPSK modulation with a square pulse shape. The information is being transmitted in the phase of the carrier. The modulated carrier can indeed be convolved with the channel's impulse response; however, please see Matt's comment to your question first.
– MBaz
Jun 26, 2015 at 14:48
• Thanks I edited my question. Can you explain about the demodulation process ?can I just multiply it by carrier again and take a integral and do the detection ? Jun 26, 2015 at 21:43