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Inside Audio Filtering Part 2: More Fundamental FIR Filter Concepts & Applications In Loudspeakers

Continuing the discussion by considering computational complexity and the key benefits of FIR filters.

Independent Control Of Magnitude & Phase

With most IIR filters, the phase response is inherently linked with the magnitude response. (One exception is the IIR all-pass filter.) A huge benefit of FIR filtering is the ability to manipulate magnitude and phase independently. Following are four FIR filter examples. Each have the same magnitude response but very different phase responses.

Example 1: Minimum-phase FIR filter. Previously we showed how both IIR and FIR filters use sample delays (as well as coefficients) to achieve their intended changes in the frequency response. A minimum-phase filter effects EQ while adding the least amount of delay to the audio signal. (This is one of the reasons long FIR filter-based EQ in PA systems is typically minimum phase.)

A characteristic of a minimum-phase filter is that its impulse response has larger coefficients at or near the start of the impulse response. Figure 2 shows a minimum phase FIR filter that effects a HP near 100 Hz and some EQ. While the FIR filter length is 42.7 millisecond (ms), the effective delay is negligible.

Figure 2: Above, minimum-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green); below, minimum-phase 2048 tap FIR filter frequency response (both fs = 48 kHz).

Example 2: Linear-phase FIR filter. Figure 3 shows a FIR filter with the same magnitude response but with a flat or linear phase. The bulk delay through the filter is equivalent to the peak location of the filter: here 1024 samples or 21.3 ms.

Figure 3: Above, linear-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green); below, linear-phase 2048 tap FIR filter frequency response (both fs = 48 kHz).

Example 3: Maximum-phase FIR filter. Figure 4 shows a FIR filter with the same magnitude response but with maximum-phase; this is the opposite or inverse phase of the minimum-phase filter above. The impulse response is the time reverse of the minimum-phase impulse response and so the bulk delay through the filter is approximately the length of the filter, 42.7 ms.

Figure 4: Above: maximum-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green); below, maximum-phase 2048 tap FIR filter frequency response (both fs = 48 kHz).

Example 4: Mixed-phase FIR filter. Finally (Figure 5), we have an arbitrary phase or mixed-phase FIR filter with the same frequency response. The bulk delay through the filter is approximately the location of the filter peak; here ~1480 samples or 30.4 ms. Where the peak is placed depends on the desired characteristics of the FIR filter and how those characteristics can be achieved within the tap length limit; here 2048 taps.

Figure 5: Above, mixed-phase 2048 tap FIR filter impulse response (dark green) and dB magnitude of the impulse response (light green); below, mixed-phase 2048 tap FIR filter frequency response (both fs = 48 kHz).

Why do we care about mixed-phase behavior? So we can push a loudspeaker’s phase to where we want it!

Why is independent phase useful? A loudspeaker driver can be thought of as a minimum-phase filter (when comparing the acoustic output with the electrical signal into the driver). When using minimum-phase EQ to bring a loudspeaker driver’s magnitude response closer to “flat,” the phase of the loudspeaker driver also flattens and moves closer to linear phase (at least within the audible pass band of the loudspeaker).

However, in a typical multi-way loudspeaker, the IIR HP and LP crossover filters (as well as polarity, delay and acoustic filters such as ports) all add frequency-varying extra phase. Because of this extra phase, a multi-way loudspeaker can be thought of as a minimum phase system PLUS some all-pass filters.

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