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Suppose we have a filter $L$ defined by $(L\circ x)_n=(x\ast h)_n,$ where $$h_n=\begin{cases}\frac{\sin(\frac{\pi}{6}n)+\sin(\frac{\pi}{2}n)}{\pi n},&n\neq 0,\\ 2/3,&n=0.\end{cases}$$ Here $n$ is an integer. Determine the output of this filter with the input $x_n=\sin(\frac{\pi}{8}n)-3\cos(\frac{\pi}{4}n).$

My attempt: First I tried to calculate the discrete-time Fourier transform of both $x$ and $h$, namely, $$X(e^{i\lambda})=\sum_{n\in\mathbb{Z}}\left(\sin(\frac{\pi}{8}n)-3\cos(\frac{\pi}{4}n)\right)e^{-in\lambda}$$ and $$H(e^{i\lambda})=\sum_{n\in\mathbb{Z}}\left(\frac{\sin(\frac{\pi}{6}n)+\sin(\frac{\pi}{2}n)}{\pi n}\right)e^{-in\lambda}.$$

Since the Fourier transform of the convolution is equal to the product of Fourier transforms, the DTFT of $x\ast h$ is $X(e^{i\lambda})H(e^{i\lambda}).$

However, how should I calculate this product and take the inverse transform?

Thanks in advance!

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1 Answer 1

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The DTFT and the Convolution Theorem for the DTFT are the correct tools to use.

This problem can be easily solved with table lookups, instead of direct computation. (I find the Inverse DTFT a lot easier to directly compute than the forward DTFT anyway.) Here's a convenient table Table of Discrete Time Fourier Transform Pairs . Since the DTFT is cyclic in the frequency domain, you need only concern yourself with the interval $[-\pi, \pi]$ in the frequency domain.

Note that your $h[n]$ can be written as

$$h[n] = \dfrac{1}{6}\mathrm{sinc}\left(\dfrac{n}{6}\right) + \dfrac{1}{2}\mathrm{sinc}\left(\dfrac{n}{2}\right)$$

which transforms to a sum of 2 rectangle functions of differing widths and magnitudes.

Your $x[n]$ will transform to a sum of pairs of delta functions at the 4 frequencies.

The multiplication of $H\left(e^{i\lambda}\right)X\left(e^{i\lambda}\right)$ should be simple. That product's inverse DTFT should be simple to compute directly, but may also be amenable to table lookup..

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