Note 34. Periodicities in Linear-Phase FIR Responses
Most filter design tools generate a filter’s frequency response in magnitude-phase form. When working with linear-phase filters, it is sometimes convenient to view the filters’ responses in amplitude-phase form. This note explores some of the issues that arise in converting a magnitude-phase form into an amplitude-phase form.
The magnitude-phase response can be converted into the amplitude-phase response using Recipe 34.1. However, when this is done for Type 2 or Type 4 linear-phase FIR filters, the resulting normalized-frequency amplitude response is periodic in 4π. This result appears to contradict the fact that the normalized-frequency response for any digital filter should be periodic in 2π. This note explores this apparent contradiction.
Example 34.1. Responses for Type 1 Filter
Figure 34.1 shows the magnitude and phase responses for a 7-tap FIR “boxcar” averaging filter, which happens to be a Type 1 linear-phase filter. The impulse response of this filter is rectangular, having equal weights for all coefficients.
Figure 34.1. Magnitude and phase responses for a 7-tap “boxcar” FIR averaging filter
The nickname “boxcar” comes from the appearance of the rectangular impulse response as it slides along the time axis. The nulls in the magnitude response occur at the same frequencies at which the phase response exhibits jump discontinuities of 180 degrees, or π radians.
Using Recipe 34.1, the magnitude and phase responses of Figure 34.1 can be converted into the amplitude and phase responses of Figure 34.2. The zero crossings in the amplitude response correspond to the nulls in the magnitude response of Figure 34.1.
Figure 34.2. Amplitude and phase responses for a 7-tap “boxcar” FIR averaging filter
The phase response shown in Figure 34.3 has phase jumps of 3 × 360 = 1080 degrees removed at frequencies corresponding to odd multiples of π radians per second. The phase is presented in this way to show that the phase is a ramp of constant slope across the full 2π of the baseband and each of its images. The phase decreases by 1080 degrees across each image, and we’re showing a 1080-degree jump increase at the end of each image. Thus the phase starts at the same value of 540 degrees at the left-hand edge of each image. If we remove all jumps that are integer multiples of 360 degrees, the phase response is equivalent to the continuous ramp shown in Figure 34.3. Notice that the slope of this ramp agrees with Eq. (33.2) from Note 33.
Figure 34.3. Phase response equivalent to Figure 34.2 with all phase jumps of 360 degrees removed
* Due to the quantized and sampled nature of the numerically computed phase response, the jump discontinuities are rarely exactly 180 degrees. Any jump that is within a reasonable tolerance of 180 degrees should be considered a 180-degree jump. It should be noted that the phase response of a linear filter should not include any discontinuities other than 180-degree phase jumps.
† It is tempting to refer to removal of these phase jumps as “unwrapping” the phase. However, unwrapping is used to describe the process of removing jumps of 2π from phase responses. Such removals can be performed on any phase response. Removing jumps of π, as discussed in this procedure, is appropriate only when adapting the “usual” phase response to be used in conjunction with the amplitude response rather than with the magnitude response.
Example 34.2. Responses for Type 2 Filter
Figure 34.4 shows the magnitude and phase responses for an 8-tap FIR “boxcar” averaging filter, which happens to be a Type 2 linear-phase filter. The nulls in the magnitude response occur at the same frequencies at which the phase response exhibits jump discontinuities of 180 degrees, or π radians.
Figure 34.4. Amplitude and phase responses for an 8-tap “boxcar” FIR averaging filter.
Using Recipe 34.1, the magnitude and phase responses of Figure 34.4 can be converted into the amplitude and phase responses of Figure 34.5. The zero crossings in the amplitude response correspond to the nulls in the magnitude response of Figure 34.4. As shown, the amplitude response is periodic in 4π, with even images and odd images having opposite signs.
Figure 34.5. Amplitude and unwrapped phase responses for an 8-tap “boxcar” FIR averaging filter
The phase decreases by 1260 degrees across each image, but because 1260 is not an integer multiple of 360 degrees, we cannot simply insert jumps of 360n degrees at the end of each image to make the phase response periodic in 2π. The phase response can be made periodic in 4π by alternating phase jumps of 1080 degrees and 1440 degrees, as shown in Figure 34.6.
Figure 34.6. Wrapped version of phase response from Figure 34.5 showing alternating phase jumps of 1080 and 1440 degrees
Another possibility that might come to mind would be to reverse the sign of alternate images to yield the amplitude response shown in Figure 34.7, which is periodic in 2π. The phase response must be changed to compensate for the sign changes in the amplitude response. We can add 180 degrees to the 1080-degree jump and subtract 180 degrees from the 1440-degree jump to obtain a phase response that is also periodic in 2π, as shown. Unfortunately, this phase response is equivalent to the unwrapped response shown in Figure 34.8, which cannot be made into a continuous ramp.
Figure 34.7. Type 2 amplitude and phase responses forced to be periodic in 2π
Figure 34.8. Unwrapped version of phase response from Figure 34.7 showing discontinuities that cannot be removed unless the amplitude response is allowed to be periodic in 4π
The bottom line is that the normalized-frequency response for a digital filter can always be made periodic in 2π, but there may be discontinuities at the inter-image boundaries. In some cases, to obtain a response without discontinuities, it is necessary to cast the response in terms of components that are periodic in 4π rather than in the expected 2π.