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The DTFT and inverse DTFT are defined as follows: $X(\omega) ~=~ \sum_{n=-\infty}^{\infty} x[n]\, e^{-j \omega n}, ~~~~~~ -\pi \leq \omega \lt \pi$ $x[n]~=~ \int_{-\pi}^\pi X(\omega)\, e^{j\omega n} \, \frac{d\omega}{2\pi} , ~~~~ \infty\lt n\lt\infty $ Let's work out some examples of DTFT and inverse DTFT calculations
Impulse response of an ideal lowpass filter
We'll start with an ideal lowpass filter. From its frequency response we can see that it blocks all incoming frequencies having a magnitude greater than $|\omega_c|$: $H(\omega) = \begin{cases} 1&-\omega_c \leq \omega \leq \omega_c \\ 0&{\sf otherwise} \\ \end{cases}$
Dtft of a moving average system
Consider now a moving average system, where the system output at a time $n$ is the average of $M$ input values, symmetric about $n$. The impulse response of such a system is a pulse function: $p[n]= \begin{cases} 1&-M\leq n \leq M \\ 0&{\sf otherwise} \\ \end{cases}$
Dtft of a one-sided exponential
Having found the frequency response of a moving average system, let's now do the same for a recursive average system. The input/output relationship of such a system can be expressed as $y[n]=x[n]+\alpha y[n-1],$ where $|\alpha|\lt 1$. The impulse response for such a system is $h[n]=\alpha^n u[n]$:Read also:
OpenStax, Discrete-time signals and systems. OpenStax CNX. Oct 07, 2015 Download for free at https://legacy.cnx.org/content/col11868/1.2
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