1. We wish to extract tones at 50 and 100 Hz from tones at 1000 and 1100 Hz and then downsample the output by the largest possible factor. Assume the signal containing the four tones is sampled at 16 kHz. Use a Kaiser window FIR filter with a passband from 0 to 150Hz (1 dB down at band edge) and attenuation > 60 dB at 250 Hz. Implement the filter, downsample the output, and examine the output spectrum to ensure that it meets specifications. Now reimplement the filtering and downsampling with a single polyphase downsampling filter.
Compare the number of operations required for the original FIR implementation and the downsampling implementation.
2. Upsample the output of the downsampling filter of Question 1 back to the original 16 kHz sampling rate by designing an appropriate upsampling filter using zero packing and low-pass filtering. Then reimplement as a polyphase upsampling filter. Check the output of the
polyphase upsampler against the filtered zero-packed output to show they are the same. Compare the number of operations required for the original single-rate FIR implementation from Q1 and the polyphase downsampling/upsampling implementation in this question.
3. Without incurring additional computation, modify the polyphase upsampling filter to frequency shift the data by heterodyning with a 2 kHz carrier. That is the tones at 50, and 100 Hz will now be placed at 1900, 1950, 2050, and 2100 Hz.