Viewing Digital Filtering as a Poly-Phase Process

Practically all digital filters can be viewed as an addition or subtraction of different phase shifted versions of the input signal. Analog designs perform filter by the same theoretical idea; capacitors and inductors introduce phase changes via propagation of electric fields, which are added together to filter the input signal.

In digital signal processing, one of the simplest filters that can be deployed is a first order finite impulse response (FIR) low pass filter (LPF). This simple digital filter example shows how filtering can be viewed and implemented in a poly-phase structure.

Equation 1.2 is the causal transfer function of the LPF, defined as the z-transform of the output divided by the z-transform of the input given in Equation 1.1.

(1.1)

(1.2)

Solving Equation 1.1 for Y(z) and taking the inverse z-transform produces the following time domain input-output equation in Equation 1.3.

(1.3)
where x(n) denotes the current input and x(n-1) denotes the previous input.

Figure 1.1 shows a block diagram that can represent the input output equation in Equation 1.3.

LPF polyphase

Figure 1.1 Simple LPF Block Diagram

The output is an addition of two different phases of the input signal. Let us now view the effects of each phase. The element in Path 2 is a unit delay. The phase response of a unit delay is displayed in Figure 1.2.

Phase Response of Simple Delay

Figure 1.2 Phase Response of a Unit Delay

The unit delay is a linear phase element; that is to say, the phase response is linear across normalized frequency. At a normalized frequency of zero, there is zero degrees phase change; at a normalized frequency of one, one times pi radians also known as the Nyquist frequency, there is a 180-degree phase change.

Path 1 of Figure 1.1 contains only a simple multiply. The phase response of a simple multiply is shown in Figure 1.3. The simple multiply does not change the phase of the input. When these two paths with different phase responses are added, phase cancellation occurs as shown in Figure 1.4. Each frequency of the input signal will be affected by a different amount of phase cancellation. Zero degrees of phase cancellation do not affect the input signal; the magnitude and phase of the output signal are identical to the input signal. 180 degrees of phase cancellation results in complete signal cancellation; the magnitude is zero and the output phase would be 90 degrees if there were a signal with which to represent the phase. Figure 1.5 shows a time domain representation of the filtering effects at normalized frequencies of zero, 0.5, and one. Figure 1.6 shows the overall magnitude and phase response of the LPF system. Figure 1.7 shows the pole/zero plot for the simple LPF. There is a zero at location (-1,0) that can be found by setting the numerator of Equation 1.2 equal to zero. This zero, which is located on the unit circle, corresponds exactly to the 180-degree phase cancellation in the system.

Phase Response of path1

Figure 1.3 Phase Response of a Simple Multiply

phase change graph copy

Figure 1.4 Phase Cancellation in the Simple LPF

phase cancellation output

Figure 1.5 Filtering Effects at Three Distinct Frequencies

simple LPF mag and phase

Figure 1.6 Magnitude and Phase Response of the Simple LPF

simple LPF PZ plot

Figure 1.7 Pole/Zero Plot of the Simple LPF

N-path_polyphase

Figure 1.8 N-Path Poly-phase Filter

The simple LPF can be viewed as a combination of many phases, or poly-phase, that result in a filter operation. This poly-phase view is easy to visualize and understand in a simple system, but can be extended to a more complex system. Figure 1.8 shows how an FIR filter of any length can be expressed in poly-phase format. The phase cancellation of all of the paths can be designed to obtain the desired filtering operation. In the FIR poly-phase structure, the change of phase in each path is constrained to be linear phase from zero to 180 degrees and unity magnitude due to the nature of the unit delay element. However, the phase changing element does not have to be a single unit delay element. The unit delay may be replaced with a structure with non-linear phase, and a more robust and efficient poly-phase system can be created. For example, structures that result in infinite impulse response (IIR) through signal feedback could be used in each path to affect the phase.