Software Defined Radio Lec 5 – Multi-Rate DSP 2

1. Software Defined Radio Lec5 – Multi-Rate DSP 2 SajjadHussain, MCS-NUST

2. Outline for Today’s Lecture • Multi-Rate Filtering • Introduction • Principles of Downsampling -Decimation • Principles of Upsampling – Interpolation • Zero Insertion • Zero-Order Hold • Multi-Rate Identities – Noble Identities • Multi-Stage Structures • CIC Filters • Polyphase filters • Digital Filter banks

3. Polyphase filters • One of the principal benefits of the implementation of multi-rate systems is the large reduction in the computational cost •  use of inexpensive, low-speed processing elements in place of costly high speed elements

4. Polyphase Interpolation Filters • The computational efficiency of the Interpolator filter structure can also be achieved by reducing the large FIR filter of length K into a set of smaller filters of length L. • These smaller filters will have a length N = K/L, where K is selected to be a multiple of L. • Because the interpolation process inserts L - 1 zeros between successive values of x(n), only N out of the K input values stored in the FIR filter at any one time are nonzero. • At one time instant, these nonzero values coincide and are multiplied by the filter coefficients h(0), h(L), h(2L), ... , h(K - L). • In the following instant, the nonzero values of the input sequence coincide and are multiplied by the filter coefficients h(1), h(L + 1), h(2L + 1), ..., h(K - L + 1), and so on. • This observation leads us to define a set of smaller filters called polyphase filters, with unit sample responses: • The polyphase filter can also be viewed as a set of L subfilters connected to a common delay line.

5. Polyphase Interpolation - Background

6. Polyphase Interpolation

7. Polyphase Interpolation

8. Polyphase Interpolation • For the analysis we assume that hI(m) is a finite impulse response (FIR) filter with length N • In the above figure for each new sample x(n), the interpolator gives I output samples of y(m) • Upper path (k=0) gives y(m) for m=0, ±I, ±2I,… • Next path (k=1) gives y(m) for 1, ±(I+1), (2I+1)…

9. Polyphase Decimation • The decimation process can be decomposed into a sum of D parallel filtering stages, where the filters pk(m) are formed by decimating hD(m+k) by a factor D. • Intuitively, it is apparent that as a sample progresses through h(n), only a subset of filter coefficients (one out of D) will engage the sample to lead to an output value. • The set of coefficients that engage that sample represent a single branch

10. Polyphase Decimation

11. Polyphase Decimation - Background

12. Polyphase Decimation - Background

13. Polyphase Decimation

14. Polyphase Decimation - Example

15. Polyphase Decimation - Example

16. Digital Filter Banks • An approach for reduction of computational complexity  breaking a wideband signal into multiple bands (channels) • Each of these channels can be processed independently – low-rate processing • Other appllications  FDM to TDM

17. Digital Filter Banks • Analysis example – DFT as a Filter bank • Assumptions : • Uniform filter bank : all the channels have same BW and sampling rates • Channels outputs are critically sampled : total samples from all channel outputs together are same as total samples at input per unit time

18. Implementation of Digital Filter Banks • Implementation of DFBs follow two basic architectures : • Analyzer : for decomposing input signal into several channels • Synthesizer : for reconstructing the input by combining channel signals