chapter 8 dsp algorithm implementation wang weilian [email protected] school of information science...

CHAPTER 8 DSP Algorithm Implementation

CHAPTER 8 DSP Algorithm Implementation

Wang Weilian

[email protected]

School of Information Science and Technology

Yunnan University

OUTLINEOUTLINE

• Concepts Relating to DSP Algorithm

– Matrix Representation

– Precedence Graph

– Structure Verification

• Structure Simulation and Verification Using MATLAB

• Computation of DFT

– Goertzel’s Algorithm

– FFT

– Inverse DFT computation

• Summary

MATRIX REPRESENTATION OF THE DIGITAL FILTER STRUCTURE


• Consider the structure figure:

• The structure can be described by a set of equations:

1 5

2 1 3

13 2

4 3 2

15 4

1 5

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

W z X z W z

W z W z W z

W z z W z

W z W z W z

W z z W z

Y z W z W z



• In the time-domain, the equivalent set of equations is as follows:

• The above set of equations are non-computable in the present order.



• Reorder the above equations as follows:

• The above ordered set of equations can be implemented in the sequential order and is computable.



• A simple way to examine the computability of structure equations is by writing the equations in a matrix form.

• Define four matrices as follows:



• The first ordered set of equations can be described in the compact matrix form:

• The matrix equation is not computable because all elements of the matrix F on the diagonal and above diagonal are not all zeros.



• Computability condition: all elements of the F matrix on the diagonal and above diagonal must be zeros.

• The F matrix for the second ordered set of equations is

• It can be seen that the above F matrix satisfies the computability condition.

PRECEDENCE GRAPHPRECEDENCE GRAPH

• Precedence graph can be used to test the computability of a digital filter structure. It is developed from the signal flow-graph of the digital filter structure.

• The signal flow-graph description of

is shown below

PRECEDENCE GRAPHPRECEDENCE GRAPH

• The reduced signal flow-graph is

• The precedence graph of the above figure is

STRUCTURE VERIFICATIONSTRUCTURE VERIFICATION

• A digital filter structure is verified to ensure that no computational and / or other errors taken place and that the structure is indeed characterized by H (z).

• Consider a causal 3rd order IIR transfer function

• Its unit sample response is {h[n]}, then


• From the above equation, we arrive at

in the time-domain, the equivalent convolution sum is

for n=0,1,2,…,6, we have


• The above equations can be written in the matrix form

in the partitioned form, it can be written as


where

• Solving the second equation, we get

• Substituting above in the first equation, we arrive at

• In the case of N-th order IIR filter, the coefficients of its transfer function can be determined from the first 2N+1 impulse response samples.

Structure Simulation & Verification Using MATLAB


• As shown earlier, the set of equations describing the filter structure must be ordered properly to ensure computability.

• The procedure is to express the output of each adder and filter output variable in terms of all incoming variables.

• From the computed impulse response samples, the structure can be verified by determining the transfer function coefficients using the M-file strucver.

• The M-file filter implements IIR filter in the transposed direct form II structure shown below for a 3rd order filter.

• As indicated in the figure, d(1) has been assumed to be equal to 1.



COMPUTATION OF DFTCOMPUTATION OF DFT

GOERTZEL’S ALGORITHMGOERTZEL’S ALGORITHM

• Goertzel’s algorithm is a recursive DFT computation scheme making use of the identity

• We can write

• Define

1kNNW


then we have

• is the direct convolution of the causal sequence

with a causal sequence

[ ]ky n


• A second order realization

• According to the above figure, DFT computation equations are

FAST FOURIER TRANSFORM ALGORITHMS

FAST FOURIER TRANSFORM ALGORITHMS

• The basic idea behind the fast Fourier transform ( FFT)

– Decompose the N-point DFT computation into computations of smaller-size DFTs

– Take advantage of the periodicity and symmetry of the complex number .

kNNW

DECIMATION-IN-TIME FFT ALGORITHMDECIMATION-IN-TIME FFT ALGORITHM

• Consider a sequence of length . Using 2-band polyphase decomposition, we can express its z-transform as

[ ]x n 2uN

• Examination of the above flow-graph reveals that each stage of the DFT computation process employs the same basic computational module

• Basic computational module is called butterfly computation.


DECIMATION-IN-FREQUENCY FFT ALGORITHM

DECIMATION-IN-FREQUENCY FFT ALGORITHM

INVERSE DFT COMPUTATIONINVERSE DFT COMPUTATION

DFT AND IDFT COMPUTATION USING MATLAB

DFT AND IDFT COMPUTATION USING MATLAB

• The following functions in the MATLAB package can be used to compute DFT and IDFT:

fft(x) ifft(x) fft(x,N) ifft(x,N)

• Since vectors in MATLAB are indexed from 1 to N, the DFT and the IDFT computed in the above MATLAB functions make use of the expressions:

( 1)( 1)

1

( 1)( 1)

1

[ ] [ ] ,1

1[ ] [ ] ,1

Nn kN

n

Nn k

Nk

X k x n W k N

x n X k W n NN

SUMMARYSUMMARY

• Some common factors in the implementation of DSP algorithm

• The implementation carried out by the set of equations describing the algorithm

• The computability condition of these equations

• Fast Fourier transform (FFT)

• MATLAB-based implementation of digital filtering algorithms

chapter 8 dsp algorithm implementation wang weilian [email protected] school of information science...

Documents

structure figure

f matrix

matrix equation

digital filter structurein

digital filter structurethe

ordered set of equations

equivalent set of equations

compact matrix form