# Ideal Lowpass Filters [closed]

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!

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.)

• Well, I understood where i made it wrong. Thankyou so much for your help. Keep helping people. Jul 6 '20 at 17:30

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?

• 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. Jul 5 '20 at 19:23