Digital Signal Processing
Digital Filter Design
You design a digital filter by specifying the frequency response of the filter in the frequency domain. The specification includes the digital filter’s cutoff frequencies, passband and stopband edge frequencies, passband ripple, and stopband attenuation. Figure 8 through Figure 11 illustrate the filter specification for each type of digital filter (Oppenheim and Schafer 2010; Parks and Burrus 1987).
Figure 8: Filter Specification for Lowpass Filter
Figure 9: Filter Specification for Highpass Filter
Figure 10: Filter Specification for Bandpass Filter
Figure 11: Filter Specification for Bandstop Filter
A digital filter’s cutoff frequency, 
, is defined as the frequency at which the power of the frequency response reaches half the unity power, or equivalently 
of the unity magnitude, which is approximately 
= 3, measured in decibels (dB). Because half power is about 3 dB away from unity power, this frequency is often called the 3 dB cutoff frequency. For lowpass and highpass filters, you need to specify only one cutoff frequency value. However, for bandpass and bandstop filters, you need to specify two cutoff frequency values, as illustrated in Figure 10 and Figure 11.
The digital filter passband and stopband edge frequencies are 
and 
, respectively. For lowpass and highpass filters, you need only one value of each edge frequency. However, for bandpass and bandstop filters, you need two values of the passband edge frequencies (
and 
) and two values of the stopband edge frequencies (
and 
). The frequency range from the passband edge frequency to the stopband edge frequency is the transition band. The transition band has a frequency response that is unspecified (Oppenheim and Schafer 2010; Parks and Burrus 1987; Roy 2005; Constantinides 1970).
The digital filter passband and stopband can contain oscillations known as ripples. The symbol 
represents the magnitude of the passband ripple, which equals the maximum deviation from the unity magnitude. The symbol 
represents the magnitude response of the stopband attenuation, which equals the maximum deviation from zero. The passband ripple (
) and stopband attenuation (
) are usually measured in decibels and defined, respectively, as
(dB) 
(dB)
where the absolute value of the passband ripple, , must be less than the absolute value of the stopband attenuation, ; that is, 
.
The values of the passband and stopband edge frequencies are normalized values between 0 and 1, exclusive, where 1 corresponds to the normalized Nyquist frequency (
rad/sample). When you specify the passband and stopband edge frequencies, the following expressions must be satisfied:
Lowpass filter:
Highpass filter:
Bandpass filter:
Bandstop filter:
