# Frequency Response Problem for given system

ace130
Homework Statement:
See System equation
Relevant Equations:
N/A
Hello,
I am given an exercise in which I need to answer some questions, for this given system : y(n)=1/3[x(n)+x(n-1)]

1/ find the impulse response h(n), then H(n)
2/ calculate the magnitude ||H(n)|| and tell the nature of the filter
3/ calculate Fc which is cutoff frequency when Fe=8Khz , and then calculate Fa at 10 dB of Attenuation
4/ find y(n) if the inpux x(n)=cos(2*pi*f*n)

My attempt was the following
- Applying a Dirac δ(n) as impulse I get h(n)=1/3[δ(n)+δ(n-1)]
- Using Fourier transform for H(n)=FT[h(n)] I get H(n)=2/3[1+exp(2*pi*j*f)] ;
- Using trigonometry formula Cos(x)=1/2(e^jx+e^-jx) I found the magnitude ||H(n)|| = |cos(pi*j*f)|

BUT I still cannot understand what is cuttof frequency, or how to calculate it. And what is the meaning of 10 dB of Attenuation and How to calculate Fa.

Master1022
Hi,

I'll have a go at answering this. You might have more luck putting this in the Engineering forum, as this seems like a signal processing question (correct me if I am wrong). If this is a signal processing course, have you learned about concepts like: z-transform, high pass/low pass/band pass filters, Bode Plots?

Hello,
I am given an exercise in which I need to answer some questions, for this given system : y(n)=1/3[x(n)+x(n-1)]

1/ find the impulse response h(n), then H(n)
I agree with your answer for the discrete impulse response ##h(n)##. To double check this, you could take the Z-transform of the equation, re-arrange to get the transfer function ## H(z) = \frac{Y(z)}{X(z)} ## and then inverse z-transform that to get back to ##h(n)##

For ##H(n)##, I am not sure what the notation means. Judging from your answer, is that the frequency response of the impulse response (i.e. transfer function). My first thought would be to use the z-transform and then make use of the mapping from ## z \rightarrow e^{sT} ## where ## s ## is the Laplace transform variable which can be written as ## z = e^{j \omega T} ## (we aren't interested in the real part of ## s ## because we are just looking at the imaginary axis in s-space). Anyways, all that text means I think you need to do:
Find H(z) --> let ## z = e^{j \omega T} ## which gives you ## H(j \omega) ##

2/ calculate the magnitude ||H(n)|| and tell the nature of the filter
So this part will just require us to use above answer. My frequency response was ## \frac{e^{j\omega T} + 1}{3 e^{j \omega T}} ##. This can be simplified using tricks similar to what you have done and you should end up with a similar answer for the magnitude. From there, I think looking at the shape should give an indication of the filter type (high pass, low pass, band pass, etc.). (can also look into FIR/IIR filters if you have learned about those)

3/ calculate Fc which is cutoff frequency when Fe=8Khz , and then calculate Fa at 10 dB of Attenuation
Sorry, can you clarify what ##F_e## is? Is that a sampling frequency?

BUT I still cannot understand what is cuttof frequency, or how to calculate it.
'Cut-off' frequencies refer to the -3dB point. They are characteristic features usually seen on Bode plots. Have you come across those before? The 3dB point represents when the POWER has gone down to 1/2 of its original value, or when the VOLTAGE has gone down to ## 1/\sqrt{2} ## of its original value.

And what is the meaning of 10 dB of Attenuation
Decibels (dB) are the units which are sometimes used to measure attenuation in systems. I would have a read about them, as it is quite a bit to explain here. In short, they are a logarithmic scale of measurement.

Hope that is of some help. For the attenuation questions, I think looking on google will help (there are lots of results that come up for those topics)