View the filter characteristics for each stage: 2:1, 4:3, 5:7, and 4:7. View the signal at the intermediate sample rates: 88.2 kHz, 117.6 kHz, 84 kHz, and 48 kHz.

Converting sampling rates from 44.1 kHz to 48 kHz is a difficult problem. A naive approach would be to interpolate (upsample) to a sampling frequency which is the least common multiple of these two frequencies, filter to prevent aliasing, then decimate (downsample) to the desired output rate. Unfortunately the sampling rate ratio in this case is 160:147. This would require interpolating to an intermediate frequency of 7.056 MHz. Designing a lowpass filter with a pass band of 0-20 kHz and a stop band of 22.05-3528 kHz would be very challenging. Such a high-Q filter would require many, many coefficients to obtain reasonable performance.

A more reasonable approach is to perform the rate conversion in multiple stages. Rate conversion ratios are chosen by examining the prime factorization of the two sampling rates. The prime factorizations of 48000 and 44100 are 2^7 x 3 x 5^3 and 2^2 x 3^2 x 5^2 x 7^2 respectively. Thus the ratio 48000:44100 is 2^5 x 5 : 3 x 7^2 or 160:147. In this example the conversion is performed in four stages --- 2:1, 4:3, 5:7, and 4:7.

The first stage requires a filter with a relatively sharp cut-off with a transition band from 20-22.05 kHz. Because of this, the ratio for this stage was chosen to be 2:1. With the smallest possible interpolation factor of 2, the cut-off frequency of 20 kHz is as high as possible with respect to the intermediate sampling rate (88.2 kHz in this case). This means that the filter for this stage will require fewer coefficients than if a higher interpolation factor had been chosen. Unfortunately, no decimation can take place in this stage since the smallest decimation factor, which is 3, would result in a loss of high-frequency energy from the original signal.

The first filter, which has 173 taps, interpolates by a factor of 2 and does not decimate. The pass band is 0-20 kHz and the stop band is 22.05-44.1 kHz. Note that the filter operates at a sampling rate of 2 x 44.1 = 88.2 kHz. The output of this filter is a signal at a 88.2 kHz sampling rate with no energy above 22.05 kHz.

The second filter, which has 31 taps, interpolates by a factor of 4 and decimates by a factor of 3. The pass band is 0-20 kHz and the stop bands are 44.1 kHz wide and are centered at multiples of 88.2 kHz (the sampling rate of the input to this stage). More specifically, the stop bands are 66.15-110.25 kHz and 154.35-176.4 kHz. Note that this filter operates at a sampling rate of 4 x 88.2 = 352.8 kHz. The output of this filter is a signal at a 117.6 kHz sampling rate.

The third filter, which has 33 taps, interpolates by a factor of 5 and decimates by a factor of 7. The pass band is 0-20 kHz and the stop bands are 44.1 kHz wide and are centered at multiples of 117.6 kHz (the sampling rate of the input to this stage). More specifically, the stop bands are 95.55-139.65 kHz and 213.15-257.25 kHz. Note that this filter operates at a sampling rate of 5 x 117.6 = 588 kHz. The output of this filter is a signal at a 84 kHz sampling rate.

The fourth filter, which has 33 taps, interpolates by a factor of 4 and decimates by a factor of 7. The pass band is 0-20 kHz and the stop bands are 44.1 kHz wide and are centered at multiples of 84 kHz (the sampling rate of the input to this stage). More specifically, the stop bands are 61.95-106.05 kHz and 145.95-168 kHz. Note that this filter operates at a sampling rate of 4 x 84 = 336 kHz. The output of this filter is a signal at a 48 kHz sampling rate.

Because the second filter has the same interpolation factor as the fourth and operates at a higher rate, it can actually use the same filter coefficients. This could be useful in an implementation where only small amounts of memory are available for storing filter coefficients.