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I am a beginner to Digital Signal Processing. I took this course of DSP from Coursera in which I am stuck with this question. Since, I am new, please be descriptive. Any help will be appreciated. Thanks!

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The first two lines of your solution are correct. But nothing after that is correct. (Those summation symbols should not be in your solution. Sequence y[n] is NOT equal to the sum of the samples in sequence y[n]!)

OK, for a half-band filter (whose impulse response is h[n]) here's what I have for the impulse response of your parallel-path network:

$$h_{parallel}[n] = h[n]+(-1)^nh[n]$$ $$= \frac{sin(n\pi/2)}{n\pi}+(-1)^n\frac{sin(n\pi/2)}{n\pi}$$ $$= \frac{sin(n\pi/2)}{n\pi}[1+(-1)^n]$$ $$= \frac{sin(n\pi/2)}{n\pi}[1+cos(n\pi)]$$ $$= \frac{sin(n\pi/2)}{n\pi}+\frac{sin(n\pi/2)cos(n\pi)}{n\pi}.$$ $$\text{Using a trigonometric identity, we can write:}$$ $$h_{parallel}[n]= \frac{sin(n\pi/2)}{n\pi}+\frac{sin(3n\pi/2)+sin(-n\pi/2)}{2n\pi}$$ $$= \frac{sin(n\pi/2)}{n\pi}+\frac{sin(3n\pi/2)-sin(n\pi/2)}{2n\pi}.$$ That's best I'm able to do at this point. If you evaluate that last equation for, say, n = -5,-4,-3,-2,-1,0.001,1,2,3,4,5 you will see that your parallel network's impulse response is itself an impulse sample. (Compare what I've written here to the information in my first Answer above.)

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  • $\begingroup$ Well, I understood where i made it wrong. Thankyou so much for your help. Keep helping people. $\endgroup$ Jul 6 '20 at 17:30
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This looks like a homework problem, so it wouldn't help you too much if I gave you the answer. However, the answer to your problem is actually very educational. You need to model your network using software.

Sequence $w[n]$ will be the impulse response of the lowpass half-band filter. And when you multiply a half-band filter's impulse response by $1,-1,1,-1,\ldots$ you obtain a sequence whose spectrum is equal to a highpass filter frequency response whose cutoff frequency is also $\omega_c = \pi/2$. So the combined frequency response of your parallel two-path network will be the sum of the lowpass half-band and the highpass filters' frequency responses. Devansh, what will that combined frequency response look like? So now the final question is: What is the time-domain sequence (the impulse response) whose spectrum is the sum of the lowpass half-band and the highpass filters' frequency responses?

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  • $\begingroup$ Regards Richard, Thanks for answering the question. I want to attach the solution that i made. I came up finding the final time dependent signal, which sums up to zero. So, in my opinion, answer should have been 0. But, it is wrong. Please tell me where I am doing wrong. Here is the link- drive.google.com/file/d/1HGlsXzGG3t-Ko3UDwFPkSmaIMW_vLXUR/… Thankyou. $\endgroup$ Jul 5 '20 at 19:23

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