# 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.

## Answers and Replies

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)