# $H(e^{j\omega})$ represented with a limited no of samples

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:

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

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

• What have you tried so far? The first thing I'd try (without guarantee of success) is the property of the Fourier transform where $\mathcal F \left \lbrace h[n] g[n] \right \rbrace$ = $\mathcal F \left \lbrace h[n] \right \rbrace \star \mathcal F \left \lbrace g[n] \right \rbrace$. Sep 19, 2021 at 19:06
• You need to learn about the various windows and what effect their application in the discrete-time domain has in the frequency domain. Sep 19, 2021 at 19:22

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.