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I have a low pass filter defined as:

$$ H_{lp}(e^{j\omega})=\left\{ \begin{aligned} &1 &|\omega|>\omega_c \\ &0 &\omega_c<|\omega|<\pi \end{aligned} \right. $$

and the corresponding sequence in the discrete-time domain is

$$h[n]=\frac{1}{2\pi} \int^{\omega_c}_{-\omega_c}e^{j\omega n}d\omega=\frac{\sin(\omega_cn)}{\pi n}$$

The discrete time-domain signal $h[n]$ has infinite samples.

If we consider only a finite amount of samples:

$$H_M(e^{j\omega})=\sum^{M}_{n=-M}\frac{\sin(\omega_c n)}{\pi n}e^{-j\omega n}$$

And then the book says

$H_M(e^{j\omega})$ can also be represented as:

$$H_M(e^{j\omega})=\frac{1}{2\pi}\int^{\omega_c}_{-\omega_c} \frac{\sin[(2M+1)\frac{(\omega-\theta)}{2}]}{\sin[\frac{(\omega-\theta)}{2}]} d\theta$$

How was this expression evaluated?

This is an example from Oppenheim Discrete-Time Signal Processing book

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  • $\begingroup$ What have you tried so far? The first thing I'd try (without guarantee of success) is the property of the Fourier transform where $\mathcal F \left \lbrace h[n] g[n] \right \rbrace$ = $\mathcal F \left \lbrace h[n] \right \rbrace \star \mathcal F \left \lbrace g[n] \right \rbrace$. $\endgroup$
    – TimWescott
    Sep 19 '21 at 19:06
  • $\begingroup$ You need to learn about the various windows and what effect their application in the discrete-time domain has in the frequency domain. $\endgroup$ Sep 19 '21 at 19:22
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HINT:

  1. figure out the frequency response $W(e^{j\omega})$ of a symmetric rectangular window $w[n]$ of length $2M+1$
  2. note that the truncated impulse response is $h_M[n]=h[n]w[n]$
  3. from the previous equation, note that the frequency response $H_M(e^{j\omega})$ can be written as the convolution of $W(e^{j\omega})$ with a rectangle. This results in the representation given in your question.
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  • $\begingroup$ you mean to say .... to say treat this like the signal in the time domain only multiplied by a rectangular pulse of length 2M+1...and since it is multiplied in time domain...it is convolved in frequency domain... $\endgroup$
    – Orpheus
    Sep 20 '21 at 2:33

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