We know that the frequency response of a system is the fourier transform of the impulse response. When we pass a signal through such a system, we get a product the the frequency responses in the frequency domain. In other words, convolution in the time domain corresponds to multiplication in the frequency domain:

$$ y(n) = (x*h)(n) \ftp Y(\omega) = X(\omega)H(\omega) $$

This tells us the the frequency response of the system can be described as a ratio of the input and output signal frequency responses:

$$ H(\omega) = \frac{Y(\omega)}{X(\omega)} $$


\begin{example}
Find the frequency response of the causal linear constant coefficient difference equation (LCCDE) given by
$$ \sum \limits_{k=0}^{N}a_ky(n-k) = \sum \limits_{\ell=0}^{M}b_\ell x(n-\ell) $$

We can use the following properties of the DTFT:

\begin{align*}
x(n) &\ftp X(\omega) \\
x(n-\ell) &\ftp e^{-i\ell\omega}X(\omega) \\
y(n-k) &\ftp e^{-i k \omega}Y(\omega) \\
\end{align*}

Now we can solve the LCCDE:

\begin{align*}
\sum \limits_{k=0}^{N}a_ky(n-k) &= \sum \limits_{\ell=0}^{M}b_\ell x(n-\ell) \\
\sum \limits_{k=0}^{N}a_ke^{-i k \omega}Y(\omega) &= \sum \limits_{\ell=0}^{M}b_\ell e^{-i \ell \omega}X(\omega) \\
Y(\omega) \lr{\sum \limits_{k=0}^{N}a_ke^{-i k \omega}} &= X(\omega) \lr{\sum \limits_{\ell=0}^{M}b_\ell e^{-i \ell \omega}} \\
\frac{Y(\omega)}{X(\omega)} &= \frac{\sum \limits_{\ell=0}^{M}b_\ell e^{-i \ell \omega}}{\sum \limits_{k=0}^{N}a_ke^{-i k \omega}} \\
H(\omega) &= \frac{\sum \limits_{\ell=0}^{M}b_\ell e^{-i \ell \omega}}{\sum \limits_{k=0}^{N}a_ke^{-i k \omega}} \\
\end{align*}

It turns out that $H(\omega)$ is a ratio of polynomials in $e^{i\omega}$. This is a characteristic of systems that have LCCDEs. Note that if we define $z = e^{i\omega}$, we can write this expression as 

$$ \hat{H}(z) = \frac{\sum \limits_{\ell=0}^{M}b_\ell z^{-\ell}}{\sum \limits_{k=0}^{N}a_k z^{-k}} = \frac{P(z)}{Q(z)}  $$

\end{example}

\begin{example}
What is the LCCDE of the frequency response given by $H(\omega) = \frac{1}{1-\alpha e^{-i\omega}}$?

\begin{align*}
H(\omega) &= \frac{1}{1-\alpha e^{-i\omega}} \\
\frac{Y(\omega)}{X(\omega)} &= \frac{1}{1-\alpha e^{-i\omega}} \\
Y(\omega) - \alpha e^{-i\omega}Y(\omega) &= X(\omega) \\
y(n) - \alpha y(n-1) &= x(n) \quad \mbox{inverse fourier transform of both sides} \\
\end{align*}

\end{example}


Causal LCCDEs