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)