6.5. Linear-phase Filter#
A linear-phase filter is an LTI system that has frequency response of the form:
\[\begin{equation*} H(e^{j\hat\omega}) = |H(e^{j\hat\omega})| e^{-j\hat\omega \alpha}. \end{equation*}\]That is, its phase response \(\angle H(e^{j\hat\omega}) = - \hat\omega \alpha\) is linear in \(\hat\omega\) with slope \(-\alpha\).
It is easy to check that both the phase delay and group delay of a linear-phase filter are equal to the constant \(\alpha\), i.e., \(\tau_p(e^{j\hat\omega}) = \tau_g(e^{j\hat\omega}) = \alpha\). This means that all frequency components as well as the envelope of the input signal are delayed by the same amount (\(\alpha\)). This is a desirable property in some applications, e.g., digital communications, where the envelope is employed to convey information.
6.5.1. Ideal Filter#
A linear-phase filter whose magnitude response equals \(1\) over the passband and \(0\) over the stopband is considered as an ideal filter. For example, the frequency response of an ideal lowpass filter is
(6.7)#\[\begin{split}\begin{equation} H_{\text{LP}}(e^{j\hat\omega}) = \begin{cases} e^{-j\hat\omega\alpha} & \text{if } |\hat\omega|\leq \hat\omega_{\text{LP}} \\ 0 & \text{if } \hat\omega_{\text{LP}} < |\hat\omega| \leq \pi \end{cases} \end{equation}\end{split}\]where \(\hat\omega_{\text{LP}}\) is the cutoff frequency of the ideal lowpass filter. For \(\hat\omega_{\text{LP}} = \pi\), the filter is called an ideal allpass filter.
It is easy to check that the impulse response of the ideal lowpass filter in (6.7) is
\[\begin{equation*} h_{\text{LP}}[n] = \frac{\sin \left(\hat\omega_{\text{LP}} (n - \alpha) \right)}{\pi (n - \alpha)}. \end{equation*}\]For any input signal \(x[n]\) to the ideal lowpass filter in (6.7), the output is
\[\begin{equation*} y[n] = x[n]*h_{\text{LP}}[n] = \sum_{k=-\infty}^{\infty} x[k] \, \frac{\sin \left(\hat\omega_{\text{LP}} (n - k - \alpha) \right)}{\pi (n - k - \alpha)}. \end{equation*}\]For the special case of the ideal allpass filter (\(\hat\omega_{\text{LP}} = \pi\)), if one interprets \(x[n]\) as an oversampled version of a continuous-time signal \(x(t)\) at the sampling rate \(f_s\), then \(y[n]\) is the sampled version of \(x\left(t-\frac{\alpha}{f_s}\right)\) obtained at the same sampling rate.
The frequency responses of ideal highpass, bandpass, and bandstop filters are respectively given as below:
\[\begin{align*} H_{\text{HP}}(e^{j\hat\omega}) &= \begin{cases} e^{-j\hat\omega\alpha} & \text{if } \hat\omega_{\text{LP}} < |\hat\omega| \leq \pi \\ 0 & \text{if } |\hat\omega| \leq \hat\omega_{\text{HP}} \end{cases} \\ H_{\text{BP}}(e^{j\hat\omega}) &= \begin{cases} e^{-j\hat\omega\alpha} & \text{if } \hat\omega_{l} \leq |\hat\omega| \leq \hat\omega_{u} \\ 0 & \text{if } 0 \leq |\hat\omega| < \hat\omega_{l} \text{ or } \hat\omega_{u} < |\hat\omega| \leq \pi \end{cases} \\ H_{\text{BS}}(e^{j\hat\omega}) &= \begin{cases} e^{-j\hat\omega\alpha} & \text{if } 0 \leq |\hat\omega| < \hat\omega_{l} \text{ or } \hat\omega_{u} < |\hat\omega| \leq \pi \\ 0 & \text{if } \hat\omega_{l} \leq |\hat\omega| \leq \hat\omega_{u} \end{cases} \end{align*}\]Caution
The transfer function of a general ideal filter is not rational. Hence, we can’t use an FIR/IIR filter to implement an ideal filter, except for the special case of the ideal allpass filter (\(\hat\omega_{\text{LP}} = \pi\)) with an integer \(\alpha\) where impulse response reduces to \( h_{\text{LP}}[n] = \delta[n - \alpha]\).
6.5.2. Generalized Linear-phase Filter#
A generalized linear-phase filter has frequency response of the following form:
\[\begin{equation*} H(e^{j\hat\omega}) = A(e^{j\hat\omega}) e^{-j(\hat\omega \alpha + \beta)} \end{equation*}\]where \(A(e^{j\hat\omega})\) is a real-valued (periodic) function of \(\hat\omega\).
The magnitude response of the generalized linear-phase filter is \(|H(e^{j\hat\omega})| = |A(e^{j\hat\omega})|\) and the phase response is \(\displaystyle \angle H(e^{j\hat\omega}) = \begin{cases} -\hat\omega \alpha- \beta & \text{if } A(e^{j\hat\omega}) \geq 0 \\ -\hat\omega \alpha- \beta + \pi & \text{if } A(e^{j\hat\omega}) < 0. \end{cases}\) Therefore, the group delay \(\tau_g(e^{j\hat\omega})\) is essentially (except at frequencies where \(A(e^{j\hat\omega})\) changes sign) constant at the value \(\alpha\). Hence, the constant group delay advantage of a linear-phase filter still holds for the generalized linear-phase filter.
6.5.3. Causal Generalized Linear-phase FIR Filter#
Notation
Let \(h[n]\) be the impulse response of a causal FIR filter of order \(M\) with real-valued filter taps. We say that \(h[n]\) is symmetric (antisymmetric) if \(h[n] = h[M-n]\) (\(h[n] = -h[M-n]\)) for \(n=0,1,\ldots, M\).
The following four conditions give causal generalized linear-phase FIR filters:
Symmetric \(h[n]\) with an even order \(M\):
\[\begin{equation*} H(e^{j\hat\omega}) = \underbrace{\sum_{k=0}^{\frac{M}{2}} \tilde{b}_k \cos (\hat\omega k) }_{A(e^{j\hat\omega})} \cdot e^{-j\hat\omega \frac{M}{2}} \end{equation*}\]where \(\tilde{b}_0 = h\left[\frac{M}{2}\right]\) and \(\tilde{b}_k = 2 h\left[\frac{M}{2} -k\right]\) for \(k=1,2,\ldots, \frac{M}{2}\).
Symmetric \(h[n]\) with an odd order \(M\):
\[\begin{equation*} H(e^{j\hat\omega}) = \underbrace{\sum_{k=0}^{\frac{M+1}{2}} \tilde{b}_k \cos \left(\hat\omega (k - \frac{1}{2}) \right)}_{A(e^{j\hat\omega})} \cdot e^{-j\hat\omega \frac{M}{2}} \end{equation*}\]where \(\tilde{b}_k = 2 h\left[\frac{M+1}{2} -k\right]\) for \(k=1,2,\ldots, \frac{M+1}{2}\).
Antisymmetric \(h[n]\) with an even order \(M\):
\[\begin{equation*} H(e^{j\hat\omega}) = \underbrace{\sum_{k=0}^{\frac{M}{2}} \tilde{b}_k \sin (\hat\omega k)}_{A(e^{j\hat\omega})} \cdot e^{-j\left(\hat\omega \frac{M}{2} - \frac{\pi}{2} \right)} \end{equation*}\]where \(\tilde{b}_k = 2 h\left[\frac{M}{2} -k\right]\) for \(k=1,2,\ldots, \frac{M}{2}\).
Antisymmetric \(h[n]\) with an odd order \(M\):
\[\begin{equation*} H(e^{j\hat\omega}) = \underbrace{\sum_{k=0}^{\frac{M+1}{2}} \tilde{b}_k \sin \left(\hat\omega (k - \frac{1}{2}) \right)}_{A(e^{j\hat\omega})} \cdot e^{-j\left(\hat\omega \frac{M}{2} - \frac{\pi}{2} \right)} \end{equation*}\]where \(\tilde{b}_k = 2 h\left[\frac{M+1}{2} -k\right]\) for \(k=1,2,\ldots, \frac{M+1}{2}\).
The zeros of the 4 types of causal generalized linear-phase FIR filters above satisfy the following properties:
Types 1 & 2:
Since \(h[n]\) is symmetric, we have \(H(z) = z^{-M} H(z^{-1})\). Hence, if \(z_0\) is a zero of \(H(z)\), then \(\frac{1}{z_0}\) is also a zero. That is, \(H(z)\) always have zero-pairs \((z_0, \frac{1}{z_0})\). In particular, for type 2 (\(M\) is odd), \(H(-1) = -H(-1)\) which implies \(H(-1)=0\); so there must be a zero at \(z=-1\), i.e., the filter can’t be a highpass one.
Types 3 & 4:
Since \(h[n]\) is antisymmetric, we have \(H(z) = -z^{-M} H(z^{-1})\). Again, \(H(z)\) always have zero-pairs \((z_0, \frac{1}{z_0})\). In addition, \(H(1)=-H(1)\) which implies \(H(1)=0\); so there must be a zero at \(z=1\), i.e., the filter can’t be a lowpass one. For type 4 (\(M\) is odd), \(H(-1) = -H(-1)\) which implies \(H(-1)=0\); so there must be a zero at \(z=-1\), i.e., the filter can’t be a highpass one either.
It is not hard to show that any causal, generalized linear-phase, \(M\)-order FIR filter of any of the 4 types above has the following factorization:
\[\begin{equation*} H(z) = H_{\min}(z) \cdot H_{\text{uc}}(z) \cdot H_{\max}(z) \end{equation*}\]where \(H_{\text{uc}}(z)\) is the transfer function whose zeros lie solely on the unit circle, and \(H_{\min}(z)\) and \(H_{\max}(z)\) are a pair of minimum-phase and maximum-phase FIR filters of order \(\tilde{M}\) satisfying \(H_{\max}(z) = z^{-\tilde{M}} H_{\min}(z^{-1})\).
6.5.4. Frequency Transformation#
Suppose that we have a prototype design of a lowpass generalized linear-phase filter with real-valued impulse response \(h[n]\) and frequency response \(H(e^{j\hat\omega}) = A(e^{j\hat\omega}) e^{-j(\hat\omega\alpha+\beta)}\). By the conjugation property of DTFT, \(H(e^{j\hat\omega})\) must be conjugate symmetric, i.e., \(H(e^{-j\hat\omega}) = H^*(e^{j\hat\omega})\), which implies \(A(e^{-j\hat\omega}) = A(e^{j\hat\omega}) e^{2j\beta}\). It is then easy to check that only the following two cases are possible:
\(A(e^{-j\hat\omega}) = A(e^{j\hat\omega})\) and \(\beta=0\), or
\(A(e^{-j\hat\omega}) = -A(e^{j\hat\omega})\) and \(\beta=\pm \frac{\pi}{2}\).
Note that the type-1 and -2 generalized linear-phase FIR filters follow the former case while the type-3 and -4 filters follow the latter case.
Since \(|H(e^{-j\hat\omega})| = |H(e^{j\hat\omega})|\), it suffices to describe the specification of the lowpass generalized linear-phase filter over the positive frequency axis. Let its passband be \([0, \hat\omega_p]\) and stopband be \([\hat\omega_s, \pi]\), where \(\hat\omega_p \leq \hat\omega_s\). The frequency range \([\hat\omega_p, \hat\omega_s]\) is called the transition band.
We may obtain a highpass, bandpass, or bandstop filter from the lowpass prototype while maintaining the design specifications in the passband and stopband by doing some simple (circular) frequency shifting operations.
To get a highpass filter with frequency response \(\hat H(e^{j\hat\omega})\), frequency shift \(H(e^{j\hat\omega})\) by \(\pi\), i.e.,
(6.8)#\[\begin{equation} \hat H(e^{j\hat\omega}) = H(e^{j(\hat\omega-\pi)}) = A(e^{j(\hat\omega-\pi)}) e^{-j(\hat\omega\alpha+\beta - \alpha\pi)}. \end{equation}\]It is clear from (6.8) that \(\hat H(e^{j\hat\omega})\) is a generalized linear-phase filter. It is also easy to check that the passband of \(\hat H(e^{j\hat\omega})\) is \([\pi - \hat\omega_p, \pi]\), the stopband of \(\hat H(e^{j\hat\omega})\) is \([0, \pi - \hat\omega_s]\), and the specifications the passband and stopband of \(H(e^{j\hat\omega})\) carry over to those of \(\hat H(e^{j\hat\omega})\). Note that the transfer function of the transformed highpass filter \(\hat H(z) = H(-z)\), where \(H(z)\) is the transfer function of the prototype lowpass filter. This implies the highpass filter’s impulse response \(\hat h[n] = (-1)^n h[n]\).
To get a bandpass filter frequency response \(\tilde H(e^{j\hat\omega})\), frequency modulate \(h[n]\) by \(\hat\omega_0\) where \(\hat\omega_s < \hat\omega_0 \leq \pi-\hat\omega_s\). That is, obtain the impulse response of the bandpass filter as \(\tilde{h}[n] = 2h[n]\cos(\hat\omega_0 n)\). By the frequency shifting property of DTFT, we have
(6.9)#\[\begin{split}\begin{align} \tilde H(e^{j\hat\omega}) &= H(e^{j(\hat\omega-\hat\omega_0)}) + H(e^{j(\hat\omega+\hat\omega_0)}) \\ &= A(e^{j(\hat\omega-\hat\omega_0)}) e^{-j(\hat\omega\alpha-\hat\omega_0\alpha +\beta)} + A(e^{j(\hat\omega+\hat\omega_0)}) e^{-j(\hat\omega\alpha+\hat\omega_0\alpha +\beta)} \\ &\approx \begin{cases} A(e^{j(\hat\omega-\hat\omega_0)}) e^{-j(\hat\omega\alpha-\hat\omega_0\alpha +\beta)} & \text{if } \hat\omega_0 - \hat\omega_p \leq \hat\omega \leq \hat\omega_0 + \hat\omega_p \\ A(e^{j(\hat\omega+\hat\omega_0)}) e^{-j(\hat\omega\alpha+\hat\omega_0\alpha +\beta)} & \text{if } -\hat\omega_0 - \hat\omega_p \leq \hat\omega \leq -\hat\omega_0 + \hat\omega_p \\ 0 & \text{if } |\hat\omega| \leq \hat\omega_0 - \hat\omega_s \text{ or } \hat\omega_0 + \hat\omega_s \leq |\hat\omega| \leq \pi \end{cases} \end{align}\end{split}\]where the last approximation is obtained from the assumption that \(|A(e^{j\hat\omega})| \approx 0\) in the stopband of the prototype lowpass filter, i.e., for \(\hat\omega_s \leq |\hat\omega| \leq \pi\). From (6.9), we see that the passband of \(\tilde H(e^{j\hat\omega})\) is \([\hat\omega_0 - \hat\omega_p, \hat\omega_0 + \hat\omega_p]\) and the stopband of \(\tilde H(e^{j\hat\omega})\) is \([0, \hat\omega_0 - \hat\omega_s] \cup [\hat\omega_0+\hat\omega_s, \pi]\). The specifications the passband and stopband of \(H(e^{j\hat\omega})\) also approximately carry over to those of \(\tilde H(e^{j\hat\omega})\). Although \(\tilde H(e^{j\hat\omega})\) is not a generalized linear-phase filter, we may nevertheless conclude from (6.9) that the group delay of \(\tilde H(e^{j\hat\omega})\) is approximately constant at the value \(\alpha\) within its passband.
Finally, we may obtain a bandstop filter by frequency shifting \(\tilde H(e^{j\hat\omega})\) by \(\pi\).