Filtering is the oldest and most common type of signal processing, usually in the form of frequency selectivity or phase shaping, or both. Some filter applications include (1) extracting a desired signal from other signals, (2) separating signals from noise, (3) anti-aliasing in analog-to-digital converters or smoothing in digital-to-analog converters, (4) phase equalization, and (5) limiting amplifier bandwidths for reducing noise.
Filter Types
As shown in Figs. 1-5, we can classify filters based on frequency range selectivity as: (1) low-pass filters, (2) high-pass filters, (3) band-pass filters, (4) band-stop, band-reject, or notch filters, and (5) all-pass filters used for phase shaping or equalization.
Figure 1: Low-pass filter magnitude response and symbol.
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Figure 2: High-pass filter magnitude response and symbol.
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Figure 3: Band-pass filter magnitude response and symbol.
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Figure 4: Band-stop filter magnitude response and symbol.
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Figure 5: All-pass filter magnitude response and symbol.
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Note that we can easily derive high-pass and band-pass filters from their low-pass equivalents, and thus, even though most of our examples feature low-pass filters, the concepts and ideas are applicable to the other filter types.
Ideal vs. Practical Filters
Let us consider an ideal low-pass filter, whose magnitude response is shown in Fig. 6. This ideal filter response has three properties: (1) it has a flat magnitude in the pass-band, resulting in no amplitude distortion in the signals we are passing, (2) it has a "brick wall" transition region, i.e. the transition between the pass-band and stop-band is abrupt, and (3) it has infinite rejection of out-of-band signals, i.e. zero magnitude response. These characteristics make building an ideal filter rather impractical.
Figure 6: The ideal low-pass filter magnitude response.
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Figure 7: A practical low-pass filter magnitude response.
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A real filter has the following magnitude response properties, as shown in Fig. 7:
- The pass-band could contain ripples, thus causing amplitude distortion in the signals being passed by the filter.
- There is a finite transition region between the pass-band and the stop-band.
- The rejection of stop-band (or out-of-band) signals is finite.
Magnitude and Frequency Metrics
In the design of filters, we can specify the filter specifications using following parameters, as illustrated in Fig. 8:
- DC Pass-band Gain,
- The value of the magnitude transfer function at DC or as .
- Corner Frequency,
- The corner frequency specification.
- Stop-band Frequency,
- The stop-band frequency specification.
Figure 8: Filter design specifications.
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The frequency range from to is the transition region separating the pass-band from the stop-band. Instead of individual separate metrics, another way of detailing the filter specifications is by using filter mask, also shown in Fig. 8. The filter mask is a graphical representation of the allowable values the filter magnitude response can take.
Group Delay
Aside from the filter magnitude specifications, the filter phase response is also a critical parameter, and we would like to determine the how the phase affects the overall behavior of the filter. Consider a filter with transfer function , as shown in Fig. 9.
Figure 9: A 2-port representation of a filter.
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Let us apply two sinusoids:
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(1)
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The output can then be written as:
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(2)
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Note that the sinusoid at is delayed differently from the sinusoid at , resulting in phase distortion.
Recall that , thus we can write:
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(3)
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And also:
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(4)
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Thus:
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(5)
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We can then write the output as:
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(6)
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Where:
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(7)
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Let us define phase delay, as:
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(8)
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Note that if , both sinusoids will be delayed in time by , preserving the relative delays of the input sinusoids. If , the output at will be time shifted differently than the output at , leading to phase distortion.
To avoid phase distortion, we need to set , or equivalently:
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(9)
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Thus, solving the differential equation above, we get , where is a constant.
Let us further define group delay, as:
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(10)
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Filters with , or equivalently, , are called linear phase filters, and these filters do not introduce phase distortion. Note that filters with , where is a constant, are also linear phase filters but are NOT free of phase distortion. Further note that if , then we can say that there is no signal magnitude distortion. In most cases, these ideal conditions of no phase or magnitude distortion are not exactly realizable.