Frequency response and the DTFT

A filter specification is almost always written in the frequency domain: pass these frequencies, reject those, keep the delay constant across the band. To meet a specification, an engineer has to move fluently between a filter's coefficients and its frequency response. The tool for that is the discrete-time Fourier transform (DTFT), which is the z-transform evaluated on the unit circle. This lesson develops the DTFT and its properties, shows how to read magnitude and phase responses, distinguishes phase delay from group delay, explains why linear phase is valuable and when it is not needed, and introduces two special classes of filter, all-pass and minimum-phase, that appear whenever phase matters. The APS-2 example is an equalizer that corrects the response of the weatherproof speaker.

The discrete-time Fourier transform

The DTFT expresses a sequence as a continuous sum of complex exponentials e^(jωn). It is a function of the continuous frequency variable ω and is always periodic with period 2π, because e^(j(ω + 2π)n) = e^(jωn). The inverse transform integrates over any one period. Setting z = e^(jω) in the z-transform gives the DTFT, so the DTFT exists whenever the ROC includes the unit circle, which for a causal system means stability. The frequency response of an LTI system is the DTFT of its impulse response.

Its value is easiest to see with a sinusoid. If x[n] = e^(jω₀n) enters an LTI system, the output is H(e^(jω₀))e^(jω₀n): the same exponential, scaled by a complex number. A real cosine of amplitude A therefore comes out as A|H(e^(jω₀))| cos(ω₀n + ∠H(e^(jω₀))). The magnitude response says how much each frequency is amplified, and the phase response says how much it is shifted. Plotting both from 0 to π covers everything for a real filter, because for real h[n] the magnitude is even and the phase is odd in ω.

Reading magnitude and phase

Magnitude is usually plotted in decibels, 20 log₁₀|H|, against frequency in hertz or as a fraction of the Nyquist frequency. A designer reads the pass-band ripple, the stop-band attenuation and the width of the transition band from that plot. Phase is plotted in radians or degrees and is often wrapped into −π to π, which makes it look jagged. Unwrapping, adding multiples of 2π to remove the jumps, gives a smoother curve. Jumps of exactly π that remain after unwrapping are real: they happen wherever the magnitude passes through zero and the response changes sign.

Phase delay and group delay

Two measures of delay come from the phase response. The phase delay τ_p(ω) = −∠H(e^(jω))/ω is the delay experienced by a single steady sinusoid. The group delay τ_g(ω) = −d∠H/dω is the delay of the envelope of a narrow-band signal centered at ω, such as a tone burst or a modulated carrier. When the phase is linear, both are constant and equal. When it is not, different frequencies are delayed by different amounts, which smears transients and changes waveform shapes, an effect called phase distortion.

A first-order IIR low-pass filter H(z) = (1 − a)/(1 − az⁻¹) shows the contrast. Its group delay is not constant: it is largest at DC, where it equals a/(1 − a) samples, and falls toward the Nyquist frequency. With a = 0.9 the low-frequency delay is 9 samples, while near f_s/2 it is about −0.47 samples, slightly negative. That variation is the price of the IIR filter's economy: one multiply gives a smoother that would take many taps as an FIR filter, but it does not delay all frequencies equally.

When linear phase matters

Linear phase is easy to get with FIR filters, as lesson 8 shows: any filter whose coefficients are symmetric or antisymmetric about the middle has exactly linear phase, with group delay (N − 1)/2 samples for N taps. Whether it is worth paying for depends on the application. Data receivers care a great deal, because phase distortion causes intersymbol interference; the raised-cosine filters of ELE-350 lesson 6 are linear-phase FIR filters for that reason. Measurement systems that compare waveforms, such as the echo canceller of lesson 12 or a system that measures the timing of events, need predictable delay. The ear, on the other hand, is quite tolerant of phase distortion in steady sounds, and the level meter of the APS-2 measures power, which phase does not change at all. For the A-weighting and high-pass filters, Amara therefore uses IIR filters and accepts their phase response.

All-pass filters

An all-pass filter has a magnitude response of exactly 1 at every frequency; it changes only phase. Each pole at p is paired with a zero at the reciprocal conjugate 1/p*, outside the circle, so the magnitudes of pole and zero vectors cancel everywhere on the unit circle. A first-order real all-pass has the form below. All-pass filters are used as phase equalizers, to flatten the group delay of an IIR filter where that matters, as building blocks of efficient filters and fractional delays, and in audio effects. They also explain why a filter's phase is not fixed by its magnitude: multiplying any filter by an all-pass changes its phase but not its magnitude.

Minimum phase

Many different filters share the same magnitude response. Among all causal, stable filters with a given magnitude, the one with all its zeros inside the unit circle is called minimum phase. It has the least phase lag and the least group delay at every frequency, and its energy is packed as early as possible in its impulse response. Any rational filter can be written as a minimum-phase filter times an all-pass filter, by reflecting zeros outside the circle to their reciprocal positions inside. Minimum-phase filters are important in two places in the APS-2. Their inverses are also causal and stable, because inverting turns zeros into poles, so a minimum-phase speaker model can be equalized exactly. And they give the lowest latency for a given magnitude response, which is why the output path uses minimum-phase IIR filters rather than linear-phase FIR filters wherever the waveform shape does not matter.

Equalizing the APS-2 speaker

Leo measures the weatherproof speaker in its pole housing and finds a broad dip of about 6 dB centered near 2.5 kHz, caused by the grille and the cavity behind it. Speech intelligibility depends heavily on the 1 to 4 kHz region, so Amara adds a peaking equalizer to the output path: a biquad that boosts a band around 2.5 kHz by 6 dB with a quality factor of 1.4 and leaves other frequencies alone. She computes the coefficients with the standard design equations for audio peaking filters, which place a pair of poles and a pair of zeros at the same angle, with the poles slightly closer to the circle than the zeros for a boost.

import cmath, math
fs = 48000
def peaking(f0, gain_db, Q):
    A = 10 ** (gain_db / 40); w0 = 2 * math.pi * f0 / fs; al = math.sin(w0) / (2 * Q)
    b = [1 + al * A, -2 * math.cos(w0), 1 - al * A]; a = [1 + al / A, -2 * math.cos(w0), 1 - al / A]
    return [v / a[0] for v in b], [v / a[0] for v in a]
def H(b, a, f):
    z1 = cmath.exp(-2j * math.pi * f / fs)
    return sum(bk * z1 ** k for k, bk in enumerate(b)) / sum(ak * z1 ** k for k, ak in enumerate(a))
def gd(b, a, f, df=0.5):                 # group delay in samples by differentiating the phase
    w = lambda ff: cmath.phase(H(b, a, ff))
    dphi = w(f + df) - w(f - df)
    dphi = (dphi + math.pi) % (2 * math.pi) - math.pi
    return -dphi / (2 * math.pi * 2 * df / fs)
b, a = peaking(2500, 6.0, 1.4)
print("b =", [round(v, 6) for v in b]); print("a =", [round(v, 6) for v in a])
for f in (100, 1000, 1800, 2500, 3500, 6000, 15000):
    h = H(b, a, f)
    print(f"{f:6d} Hz: {20*math.log10(abs(h)):6.2f} dB, phase {round(math.degrees(cmath.phase(h)), 2) + 0.0:7.2f} deg, group delay {gd(b, a, f):6.2f} samples")
r = abs(cmath.sqrt(a[2])); rz = abs(cmath.sqrt(b[2] / b[0]))
print(f"pole radius {r:.5f}, zero radius {rz:.5f}  (both inside: minimum phase)")
for name, aa in (("first-order low-pass a=0.9", 0.9),):
    for f in (0.001, 0.25 * fs / 2, 0.999 * fs / 2):
        print(f"{name}: group delay at {f/(fs/2):5.3f} x Nyquist = {gd([1 - aa], [1, -aa], f, 0.01):6.2f} samples")

b = [1.074807, -1.751511, 0.774866]
a = [1.0, -1.751511, 0.849673]
   100 Hz:   0.01 dB, phase    1.14 deg, group delay  -1.53 samples
  1000 Hz:   0.65 dB, phase   12.03 deg, group delay  -1.71 samples
  1800 Hz:   3.15 dB, phase   19.38 deg, group delay   0.22 samples
  2500 Hz:   6.00 dB, phase    0.00 deg, group delay   6.14 samples
  3500 Hz:   3.04 dB, phase  -19.41 deg, group delay   0.03 samples
  6000 Hz:   0.65 dB, phase  -12.08 deg, group delay  -0.32 samples
 15000 Hz:   0.04 dB, phase   -3.20 deg, group delay  -0.06 samples
pole radius 0.92178, zero radius 0.84908  (both inside: minimum phase)
first-order low-pass a=0.9: group delay at 0.000 x Nyquist =   9.00 samples
first-order low-pass a=0.9: group delay at 0.250 x Nyquist =  -0.32 samples
first-order low-pass a=0.9: group delay at 0.999 x Nyquist =  -0.47 samples

The equalizer gives exactly +6.0 dB at 2.5 kHz and returns to within a fraction of a decibel of 0 dB below 500 Hz and above about 8 kHz. Its group delay peaks at about 6 samples, 0.13 ms, at the center frequency and is slightly negative on either side, which nobody will hear. Both the poles and the zeros are inside the unit circle, so the equalizer is minimum phase, and its inverse, a 6 dB cut, would also be a stable biquad. The first-order low-pass check confirms the 9-sample group delay at DC and the small negative value near the Nyquist frequency.

Ideal filters and why they cannot be built

The ideal low-pass filter passes everything below a cutoff ω_c with unity gain and zero phase and rejects everything above. Its impulse response, found from the inverse DTFT, is a sampled sinc that extends forever in both directions. It is therefore non-causal and not absolutely summable, so it is neither realizable nor strictly stable. Every practical filter is an approximation. A filter specification states how good the approximation must be: a pass band edge ω_p with ripple δ_p, a stop band edge ω_s with attenuation δ_s, and between them a transition band where the response is allowed to do anything. Lessons 8 and 9 design FIR and IIR filters to such specifications. The narrower the transition band and the smaller the ripples, the longer or higher-order the filter.