View source for HW 3 Question 4 mlo - Rhea

[[Category:ECE438Spring2009mboutin]] 
== Part 4a ==

Given 

<math>y(n) = \frac{x(n) + x(n-1) + x(n-2)}{3} \quad (1)</math>

to find H(w), we let

<math> x(n)\,\! = \exp(jwn) \quad (2)</math> 

such that

<math>y(n)\,\! = H(w)\exp(jwn)</math>

substituting equation (1) into (2)

<math>y(n)\,\! = \frac{\exp(jwn) + \exp(jw(n-1)) + \exp(jw(n-2))}{3} \quad (3)</math>

factoring out the <math>\,\! \exp(jwn)</math> from equation (3), we obtain H(w)

<math>\,\! H(w) = \frac{1 + \exp(-jw) + \exp(-jw2)}{3}</math>

== Part 4b ==
<math>\left| H(w)\right|</math>

== Part 4c ==
To find the the overall frequency response F(w) for this system, I assumed the up/down samplers canceled each other out and so we were left with the LPF and original H(w).

Combining terms

<math>\,\! F(w) = G(w)H(w)G(w)</math>
*''<span style="color:red"> It is important to realize that upsampling is not the inverse of downsampling: when you upsample after having downsampled, you introduce zeros in the signal that were not previously there.  To undo a downsampling, you have to use an interpolator (as defined in class a couple of lectures ago). </span>''--[[User:Mboutin|Mboutin]] 11:46, 10 February 2009 (UTC)

*<span style="color:black"> Ah, gotcha, thanks! --[[User:Mlo|Mlo]] 12:03, 10 February 2009 (UTC)

Trying part 4c again

To find F(w), I stayed in the time domain and wrote out the output for an input x(n).

After the first G(w) block, my output is

<math>\,\! a(n) = x(n) * g(n) </math>

after the downsampler, we have

<math>\,\!b(n) = a(Dn)</math>

after the H(w) block we have

<math>\,\!c(n) =  b(n)*h(n)</math>

after the upsampler we have

<math>\,\!d(n) = c(n/D)</math>

after the second G(w) we have

<math>\,\!y(n) = d(n)*g(n)</math>

combining terms and using linearity, we say

<math>\,\!f(n) = g(n)*h(n/D)*g(n)</math>

and taking the DTFT of that we get

<math>\,\!F(w) = G(w)G(w)H(Dw)</math>

*''<span style="color:green"> I am not sure if i follow your logic here. Where did the a(Dn) go? If i use linearity as you suggest I do not get the same answer. </span> --[[User:Jwromine|Jwromine]] 01:02, 11 February 2009 (UTC)