6.1. Frequency Response of LTI System

• The frequency response \(H(e^{j\hat\omega})\) of an LTI system is the DTFT (if exists) of the system’s impulse response \(h[n]\), i.e., \(h[n] \stackrel{\text{DTFT}}{\longleftrightarrow} H(e^{j\hat\omega})\).

6.1.1. Properties of Frequency Response

• Let \(h[n] \stackrel{z}{\longleftrightarrow} H(z)\). If the ROC of the transfer function \(H(z)\) contains the unit circle, then \(H(e^{j\hat\omega}) = H(z) \big|_{z=e^{j\hat\omega}}\).

• The convolution properties of DTFT gives:

  If \(x[n] \stackrel{\text{DTFT}}{\longleftrightarrow} X(e^{j\hat\omega})\) and \(h[n] \stackrel{\text{DTFT}}{\longleftrightarrow} H(e^{j\hat\omega})\) are the input and output signals of the LTI system, then \(Y(e^{j\hat\omega}) = H(e^{j\hat\omega}) X(e^{j\hat\omega})\).

• For the special case \(x[n] = Ae^{j\phi} e^{j\hat\omega_0 n}\), the DTFT of the output signal of the LTI system is \(Y(e^{j\hat\omega}) = H(e^{j\hat\omega}) 2\pi Ae^{j\phi} \delta((e^{j(\hat\omega - \hat\omega_0)}) = H(e^{j\hat\omega_0}) 2\pi Ae^{j\phi} \delta(e^{j(\hat\omega - \hat\omega_0)}) \), and hence the output signal is \(y[n] = H(e^{j\hat\omega_0}) Ae^{j\phi} e^{j\hat\omega_0 n}\).

• The phase delay of the LTI system is defined as \(\displaystyle \tau_p(e^{j\hat\omega}) = -\frac{ \angle H(e^{j\hat\omega})}{\hat\omega}\), and the group delay of the LTI system is defined as \(\displaystyle \tau_g(e^{j\hat\omega}) = -\frac{ d\angle H(e^{j\hat\omega})}{d\hat\omega}\). The phase and group delays are of interest in many practical applications as we will see later. The following two examples help to illustrate the physical interpretation of the phase and group delays:

  • Example 1: Let \(\displaystyle H(e^{j\hat\omega}) = e^{-j\hat\omega \tau_p(e^{j\hat\omega})}\), i.e., \(|H(e^{j\hat\omega})| = 1\) and \(\angle H(e^{j\hat\omega}) = -\hat\omega \tau_p(e^{j\hat\omega})\). Consider the sinusoidal input \(x[n] = e^{j\hat\omega_0 n}\). Then the output is

    \[\begin{equation*} y[n] = e^{-j\hat\omega_0 \tau_p(e^{j\hat\omega_0})} e^{j\hat\omega_0 n} = e^{j\hat\omega_0( n - \tau_p(e^{j\hat\omega_0}))}. \end{equation*}\]

    As a result, we may interpret the phase delay causing a phase shift of \(-\hat\omega_0 \tau_p(e^{j\hat\omega_0})\), or equivalently a time shift of \(\tau_p(e^{j\hat\omega_0})\), to the input sinusoid at frequency \(\hat\omega_0\).

    Tip

    If \(\tau_p(e^{j\hat\omega_0})\) is not an integer, we may interpret that \(x[n]\) is obtained from oversampling (with sampling rate \(f_s\)) a continuous-time sinusoid \(x(t)\) with frequency \(\hat\omega_0 f_s\) and \(y[n]\) is the sampled (at \(f_s\)) signal obtained from the delayed version \(x(t - \frac{\tau_p(e^{j\hat\omega_0})}{f_s})\).

  • Example 2: Consider the same frequency response in Example 1, but instead with input signal \(x[n] = 2\cos(\frac{\Delta\hat\omega}{2} n) e^{j\hat\omega_0 n} = e^{j(\hat\omega_0 + \frac{\Delta\hat\omega}{2}) n} + e^{j(\hat\omega_0 - \frac{\Delta\hat\omega}{2}) n}\) where \(|\Delta\hat\omega| \ll |\hat\omega_0|\). Using the following Taylor series approximations

    \[\begin{align*} \angle H(e^{j(\hat\omega_0 + \frac{\Delta\hat\omega}{2})}) &\approx \angle H(e^{j\hat\omega_0}) - \frac{\Delta\hat\omega}{2} \tau_g(e^{j\hat\omega_0}) = - \hat\omega_0 \tau_p(e^{j\hat\omega_0}) - \frac{\Delta\hat\omega}{2} \tau_g(e^{j\hat\omega_0}) \\ \angle H(e^{j(\hat\omega_0 - \frac{\Delta\hat\omega}{2})}) &\approx \angle H(e^{j\hat\omega_0}) + \frac{\Delta\hat\omega}{2} \tau_g(e^{j\hat\omega_0}) = - \hat\omega_0 \tau_p(e^{j\hat\omega_0}) + \frac{\Delta\hat\omega}{2} \tau_g(e^{j\hat\omega_0}), \end{align*}\]

    we get the output signal

    \[\begin{align*} y[n] &= e^{j\angle H(e^{j(\hat\omega_0 + \frac{\Delta\hat\omega}{2})})} e^{j(\hat\omega_0 + \frac{\Delta\hat\omega}{2}) n} + e^{j\angle H(e^{j(\hat\omega_0 - \frac{\Delta\hat\omega}{2})})} e^{j(\hat\omega_0 - \frac{\Delta\hat\omega}{2}) n} \\ & \approx 2 \cos\left(\frac{\Delta\hat\omega}{2} (n - \tau_g(e^{j\hat\omega_0})) \right) e^{j\hat\omega_0(n - \tau_p(e^{j\hat\omega_0}) )}. \end{align*}\]

    Since \(|\Delta\hat\omega| \ll |\hat\omega_0|\), the term \(2\cos(\frac{\Delta\hat\omega}{2} n)\) in \(x[n]\) slowly varies the amplitude of the underlying sinusoid \(e^{j\hat\omega_0 n}\). As a result, \(2\cos(\frac{\Delta\hat\omega}{2} n)\) is often called the envelope of \(x[n]\). The group delay of the system thus then specifies the delay in the envelope of \(x[n]\).

6.1.2. FIR and IIR Filters

• We will focus on the design of causal LTI systems whose transfer functions are in the rational form:

  \[\begin{equation*} H(z) = \frac{B(z)}{A(z)} = \frac{\sum_{k=0}^M b_k z^{-k}}{\sum_{k=0}^N a_k z^{-k}} \end{equation*}\]

  where \(a_0 = 1\). These include both FIR and IIR filters. The \(b_k\)’s and \(a_k\)’s are often called the feedforward and feedback taps, respectively.

• Let \(z_1, z_2, \ldots, z_M\) and \(p_1, p_2, \ldots, p_N\) be the roots of \(B(z)\) and \(A(z)\), respectively. We can rewrite

  \[\begin{equation*} H(z) = b_0 \frac{\prod_{k=1}^M (1- z_k z^{-1})}{\prod_{k=1}^N (1-p_k z^{-1})}. \end{equation*}\]

  If \(\{z_k\}_{k=1}^M\) and \(\{p_k\}_{k=1}^N\) do not intersect, then \(\{z_k\}_{k=1}^M\) are zeros and \(\{p_k\}_{k=1}^N\) are poles of \(H(z)\).

  Caution

  There may be additional zeros (\(M<N\)) or poles (\(M>N\)) at \(z=0\).

• If all the poles are strictly inside the unit circle, then the system is stable, and its frequency response is given by

  (6.1)\[\begin{equation} H(e^{j\hat\omega}) = b_0 \frac{\prod_{k=1}^M (1- z_k e^{-j\hat\omega})}{\prod_{k=1}^N (1-p_k e^{-j\hat\omega})}. \end{equation}\]

• Let \(|\cdot|_{\text{dB}} = 20 \log_{10} |\cdot|\). Then (6.1) gives the following formulas for the magnitude and phase responses:

  (6.2)\[\begin{split}\begin{align*} |H(e^{j\hat\omega})|_{\text{dB}} &= |b_0|_{\text{dB}} + \sum_{k=1}^M |1- z_k e^{-j\hat\omega}|_{\text{dB}} - \sum_{k=1}^N |1- p_k e^{-j\hat\omega}|_{\text{dB}} \\ \angle H(e^{j\hat\omega}) &= \angle b_0 + \sum_{k=1}^M \angle (1- z_k e^{-j\hat\omega}) - \sum_{k=1}^N \angle (1- p_k e^{-j\hat\omega}). \end{align*}\end{split}\]

  Hence, both the magnitude and phase responses are governed by those of first-order systems with transfer functions in the form of \(1-cz^{-1}\).