CHAPTER 8 DSP Algorithm Implementation Wang Weilian School of Information Science and Technology Yunnan University

OUTLINE 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:

MATRIX REPRESENTATION OF THE DIGITAL FILTER STRUCTURE In the time-domain, the equivalent set of equations is as follows: The above set of equations are non-computable in the present order.

MATRIX REPRESENTATION OF THE DIGITAL FILTER STRUCTURE Reorder the above equations as follows: The above ordered set of equations can be implemented in the sequential order and is computable.

MATRIX REPRESENTATION OF THE DIGITAL FILTER STRUCTURE A simple way to examine the computability of structure equations is by writing the equations in a matrix form. Define four matrices as follows:

MATRIX REPRESENTATION OF THE DIGITAL FILTER STRUCTURE 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.

MATRIX REPRESENTATION OF THE DIGITAL FILTER STRUCTURE 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 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 GRAPH The reduced signal flow-graph is The precedence graph of the above figure is

STRUCTURE 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

STRUCTURE VERIFICATION From the above equation, we arrive at in the time-domain, the equivalent convolution sum is for n=0,1,2,…,6, we have

STRUCTURE VERIFICATION The above equations can be written in the matrix form in the partitioned form, it can be written as

STRUCTURE VERIFICATION 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 VERIFICATION

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.

Structure Simulation & Verification Using MATLAB

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. Structure Simulation & Verification Using MATLAB

COMPUTATION OF DFT

GOERTZEL’S ALGORITHM Goertzel’s algorithm is a recursive DFT computation scheme making use of the identity We can write Define

GOERTZEL’S ALGORITHM then we have is the direct convolution of the causal sequence with a causal sequence

GOERTZEL’S ALGORITHM A second order realization According to the above figure, DFT computation equations are

GOERTZEL’S ALGORITHM

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.

DECIMATION-IN-TIME FFT ALGORITHM Consider a sequence of length. Using 2-band polyphase decomposition, we can express its z-transform as

DECIMATION-IN-TIME FFT ALGORITHM

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-TIME FFT ALGORITHM

DECIMATION-IN-FREQUENCY FFT ALGORITHM

INVERSE DFT COMPUTATION

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:

SUMMARY 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