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One way to measure the frequency response of a filter is to input separate cosine/sine wave at the desired frequency, measure the steady state output, stack the inputs/outputs into complex arrays, and divide. A more efficient way is to use the DFT, and somehow we get away with just inputting one sinusoid.

  1. Input $x_n = A \cos(\omega_0 n)$, with $\phi = 0$
  2. Measure the steady-state output: $$ \begin{align*} y_n = A | H(e^{j \omega_0}) | \cos\left (\omega_0 n + \angle H(e^{j\omega_0}) \right ) \end{align*} $$

  3. Compute the DFTs $Y_k$ and $X_k$

  4. Record $H_{\hat{k}} = \frac{Y_{\hat{k}}}{X_{\hat{k}}}$ at $\hat{k}$ that corresponds to $\omega_0$

What I'm trying to figure out is why this works, clearly there's phase information but I'm missing something in the derivation.

$$ \begin{align*} X_{\hat{k}} &= \sum_{n=0}^{N-1} x_n e^{-j \omega_0 n} \quad \text{where } \frac{2\pi}{N} \hat{k} = \omega_0 \\ &= \sum_{n=0}^{N-1} A \cos(\omega_0 n) e^{-j \omega_0 n} \\ &= \frac{A}{2} \sum_{n=0}^{N-1} \left (e^{j\omega_0 n} + e^{-j\omega_0 n} \right ) e^{-j\omega_0 n} \\ &= \frac{A}{2} \sum_{n=0}^{N-1} 1 + e^{-j 2 \omega_0 n} \end{align*} $$

$$ \begin{align*} Y_{\hat{k}} &= \sum_{n=0}^{N-1} A \left | H(e^{j\omega_0}) \right | \cos\left (\omega_0 n + \angle H(e^{j\omega_0}) \right ) e^{-j \omega_0} \\ &= \frac{A}{2} \left | H(e^{j\omega_0}) \right | \sum_{n=0}^{N-1} \left (e^{j(\omega_0 n + \angle H(e^{j\omega_0}))} + e^{-j(\omega_0 n + \angle H(e^{j \omega_0}))} \right ) e^{-j\omega_0 n} \\ &= \frac{A}{2} \left | H(e^{j\omega_0}) \right | \sum_{n=0}^{N-1} e^{j \angle H(e^{j\omega_0})} + e^{-j \angle H(e^{j \omega_0})} e^{-j 2 \omega_0 n} \\ & \; \Big\Downarrow \; \text{?} \\ &= \frac{A}{2} \left | H(e^{j\omega_0}) \right | e^{j \angle H(e^{j\omega_0})} \sum_{n=0}^{N-1} 1 + e^{-j 2 \omega_0 n} \end{align*} $$

The idea is that the phase information comes out by dividing:

$$ \begin{align*} H_{\hat{k}} &= \frac{Y_{\hat{k}}}{X_{\hat{k}}} = \left | H(e^{j\omega_0}) \right | e^{j \angle H(e^{j\omega_0})} \end{align*} $$

What am I missing?

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    $\begingroup$ DFT outputs are complex numbers, so when expressed in polar form they contain both magnitude and phase information. The phase angle of the complex value $\frac{Y_k}{X_k}$ would correspond to the filter's response. However, this only calculates the filter's response at one frequency; it doesn't yield its overall transfer function. $\endgroup$ – Jason R Dec 17 '13 at 20:05
  • $\begingroup$ Right, I know that, but how do I show that "it works" mathematically? How do I fill in the "?" in the derivation? $\endgroup$ – toast Dec 17 '13 at 21:23
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    $\begingroup$ The sum part of exponential part of the last line of $X_k$ is equal to zero, does that help? $\endgroup$ – geometrikal Dec 18 '13 at 4:11
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    $\begingroup$ @geometrikal: The exponential term only sums to zero if $\omega_0$ is a multiple of $\frac{2\pi}{N}$. Otherwise, it doesn't go away, and you have to do some factoring to massage the expression into a Dirichlet kernel (the "periodic sinc") form. $\endgroup$ – Jason R Dec 18 '13 at 14:27
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Thanks to geometrikal and Jason R, the sum of exponentials is equal to zero when $\omega_0 = \frac{2\pi}{N} \hat{k}$ and $0 \leq \hat{k} \leq N-1$, which is obvious when you realize that it's just the sum over $2 \hat{k}$ periods of $e^{-j2\omega_0n}$, where $T = \frac{2\pi}{2\omega_0} = \frac{N \pi}{2\pi \hat{k}} = \frac{N}{2 \hat{k}}$.

For $\hat{k} > N-1$, I suppose you would have to zero pad, which is where the digital sinc Jason R. mentions comes in.

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