Filtering and Downsampling

Figure 10.5: Lowpass filtering followed by downsampling.

Because downsampling by $N$ will cause aliasing for any frequencies in the original signal above $\vert\omega\vert > \pi/N$, the input signal may need to be first lowpass filtered to prevent aliasing, as shown in Fig.10.5. Suppose we implement such an anti-aliasing lowpass filter $h(n)$ as an FIR filter of length $M$ with a cutoff frequency $\pi/N$. This is drawn in direct form in Fig.10.6.

Figure 10.6: Direct-form implementation of an FIR anti-aliasing lowpass filter followed by a downsampler.

We do not need $N-1$ out of every $N$ filter output samples due to the $N:1$ downsampler. To realize this savings, we can commute the downsampler through the adders inside the FIR filter to obtain the result shown in Fig.10.7. The multipliers are now running at $1/N$ times the sampling frequency of the input signal, $x(n)$. This reduces the computation requirements by a factor of $1/N$. The downsampler outputs are called polyphase signals. This is a summed polyphase filter bank in which each ``subphase filter'' is a constant scale factor $h(m)$.

Figure 10.7: FIR lowpass filter with downsampler commuted inside the direct-form filter.

The summed polyphase signals of Fig.10.7 can be interpreted in the following ways:

1. A ``serial to parallel conversion'' from a stream of scalar samples $x(n)$ to a sequence of length $M$ buffers every $N$ samples, followed by a dot product of each buffer with $h(0:M-1)$.

2. The overall system is equivalent to a round-robin demultiplexor, with a different gain $h(m)$ for each output, followed by an $M$-sample summer which adds the ``de-interleaved'' signals together:

Figure: Demultiplex-and-sum interpretation of the polyphase signal sum of Fig.10.7.

The polyphase processing in the anti-aliasing filter of Fig.10.7 is as follows:

• The 0th subphase signal,
  $$x(nN)\left\vert _{n=0}^{\infty}\right. = [x_0,x_N,x_{2N},\ldots],$$
  is scaled by $h(0)$.

• Subphase signal 1,
  $$x(nN-1)\left\vert _{n=0}^{\infty}\right.=[x_1,x_{N-1},x_{2N-1},\ldots]$$
  is scaled by $h(1)$,

• $\cdots$

• Subphase signal $m$,
  $$x(nN-m)\left\vert _{n=0}^{\infty}\right.=[x_m,x_{N-m},x_{2N-m},\ldots]$$
  is scaled by $h(m)$

These scaled subphase signals are summed together to form the output signal

$$y(n) = \sum_{m=0}^{N-1} h(m)x(nN-m).$$