DFT N-point circular convolution of $\cos(2 \pi n/N)$ and $\sin(2 \pi n/N)$

I am attempting to solve the below problem: I was able to calculate the N point DFT for a cosine which is similar to this (note, this is the DFT for $$\cos(2\pi n k/N)$$. So for this second image assume k in the image = 1 and m in the image =k). I used a similar method to find the $$N$$ point DFT of $$\sin(2\pi n/N)$$ with the below results (same notes on the variables as above): At this point I have $$X_1(k)$$ and $$x_2(k)$$, when I multiply these together I get a very ugly series that I can't solve. I think that I am missing something here, how should I solve this problem?

Thanks

HINT : $$\cos(2\pi n/N) = 0.5e^{j2\pi n/N}+0.5e^{-j2\pi n/N}$$. This itself is a IDFT equation, where $$X_1[k] = 0.5\delta[k-1] + 0.5\delta[k+1]$$.

Similarly for $$\sin(2\pi n/N)$$, $$X_2[k] = (0.5/j)\delta[k-1]-(0.5/j)\delta[k+1]$$.

So $$Y[k] = (0.25/j)\delta[k-1] + (0.25/j)\delta[k+1]$$, and $$y[n] = (1/2) \times (0.5/j)e^{j2\pi n/N} -(1/2) \times (0.5/j)e^{-j2\pi n/N} = (1/2) \sin(2\pi n/N)$$

To expand mathematically $$X_1[k] = \sum (0.5e^{j2\pi n/N}+0.5e^{-j2\pi n/N})e^{-j2\pi kn/N}$$ $$= \sum (0.5e^{j2\pi (1-k)n/N }) +(0.5e^{j2\pi (-1-k)n/N })$$ $$= 0.5\frac{(e^{j2\pi (1-k)N/N} - 1)}{(e^{j2\pi (1-k)/N} - 1)} + 0.5\frac{(e^{j2\pi (-1-k)N/N} - 1)}{(e^{j2\pi (-1-k)/N} - 1)}$$ $$= 0.5\frac{e^{j\pi(1-k)}\sin(\pi(1-k))}{(e^{j\pi/N(1-k)}\sin(\pi/N(1-k))} + 0.5\frac{e^{j\pi(-1-k)}\sin(\pi(-1-k))}{(e^{j\pi/N(-1-k)}\sin(\pi/N(-1-k))}$$ You can use similar steps to compute $$X_2[k]$$. You need NOT proceed to multiply from here. Look at the numerator term of each fraction. $$\sin(\pi(m-k))/\sin(\pi/N(m-k))$$ is zero when $$k \ne m$$. For $$k=m$$, even before you reach the geometric series sum, you can see all terms in summation are 1. So $$X[k] = N$$ when $$k=m$$. So $$X_1[k] = N\times 0.5\delta[k-1] + N \times 0.5\delta[k+1]$$ . Though I am bit confused about the $$N$$ factor.

• Thank you very much for your response. Apr 2 '20 at 12:03
• Thank you very much for your response. However I have a question. You stated that the sin(pi (m-k) ) term is zero for all values except for m=k. For my problem, m=1. so to me, since sin(pi*(1-k)) = 0 for any integer k, I don't see how this term can equal anything but zero? Can you elaborate :) Thank you :) Apr 2 '20 at 12:11
• For $m=1$, $\sin(\pi (1-m)) = 0$ but denominator is also $0$. You cannot have $0/0$. That is why consider $m=1$ as special case. Apr 2 '20 at 12:16
• Oh, ok. since 0/0 is indeterminate. So then I should apply something like L'Hospital's Rule (take the derivative of top and bottom and evaluate at k=1). When I do this for the x1[k] = [1/2] * [ [-pi/(-pi/N)] + [-pi/(-pi/N)] ] = [1/2] * [N+N] = N when k=1 and 0 otherwise. That seems to match the answere that you got. I will try this with the sin(...) expression as well but I think I might be heading down the right path now. Thanks a lot for your help. ) Apr 2 '20 at 12:57
• Hi, I had a follow on question. So for X1[k]=0.5δ[k−1]+0.5δ[k+1], obviously this equation only has values at k=1 and k=-1. However, whenever we evaluate IDFT it is from k = 0 to N-1. In otherwords, we never have the value k=-1. Respectfully, how do you justify having k=-1? Apr 3 '20 at 1:26