DFT product of sinusoids

Question

(From Shaums DSP outline, 2nd edition, page 248, problem 6.21)

Book says, evaluate the Sum:

$$ S = \sum^{N-1}_{n=0} \Bigg( x_1[n] \ x^{*}_2[n] \Bigg) $$

when:

$$ \begin{aligned} x_1[n] = \cos\left( \frac{2\pi n k_1}{N} \right) \\ \\ x_2[n] = \cos\left( \frac{2\pi n k_2}{N} \right) \end{aligned} $$

using Property:

$$ \sum^{N-1}_{n=0} \Bigg( x_1[n]\ x^{*}_2[n] \Bigg) = \frac{1}{N} \sum^{N-1}_{k=0} \Bigg( X_1[k]\ X_2^{*}[k]\Bigg) $$

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I start by converting cosines into complex exponentials:

$$ x_1[n] = \cos\left( \frac{2\pi n k_1}{N} \right) $$

$$ x_1[n] = 0.5e^{j2\pi n k_1 / N} + 0.5e^{-j 2\pi n k_1 / N} $$

Applying Definition of DFT to find: $$ X_1[k] = \sum_{n=0}^{N-1} \Bigg( 0.5e^{j2\pi n k_1 / N} + 0.5e^{-j 2\pi n k_1 / N} \Bigg) e^{-j2\pi nk/N} $$

$$ X_1[k] = \sum_{n=0}^{N-1} \Bigg( 0.5e^{-j(2\pi n/N)(k- k_1)} + 0.5e^{-j(2\pi n/N)(k+ k_1)} \Bigg) $$

From this I determine that sum over one period of a complex exponential is zero, except when k is selected to cancel out exponential function input to zero, such as when $k=k_1$ and $k=-k_1$. Thus:

$$ X_1[k] = \begin{cases} \frac{N}{2}&(k=k_1)\ or\ (k=-k_1) \\ 0 & else \end{cases} $$

By similar means: $$ X_2[k] = \begin{cases} \frac{N}{2}&(k=k_2)\ or\ (k=-k_2) \\ 0 & else \end{cases} $$

Here's where I get lost:

Books says, therefore:

$$ \sum^{N-1}_{n=0} x_{1}[n] x^{*}_{2}[n] = \frac{1}{N}\Bigg[ \frac{N^2}{4} + \frac{N^2}{4} \Bigg] = \frac{N}{2} $$

Where does: $$ \frac{N^2}{4} + \frac{N^2}{4} $$ come from?

Answer 1

Actually you have quite done it, but lets indicate.

When the frequencies of both sinusoidal signals $x_1[n]$ and $x_2[n]$ are the same; i.e., $k_1 = k_2 = m$, then the impulses in the corresponding N-point DFT sequences $X_1[k]$ and $X_2[k]$ will occur at the same index $k=m$ and $k=-m$ (or $N-m$) with a weight of $N/2$.

Hence multiplication of $X_1[k]$ with $X_2[k]$ will be $N/2 * N/2 + N/2 * N/2 = N^2/2$, as there are two impulses in the DFT of a cosine signal...

Finally dividing by $N$ yields the result as $N/2$ when the frequencies are the same and $0$ when not.