6.1 signal flow graph representation of linear constant- coefficient difference equations 6.2 basic structures for IIR system direct forms cascade forms parallel forms lattice structure 6.3 basic structures for FIR system direct forms cascade forms structures for linear-phase FIR system lattice structure CHAPTER 6 structures for discrete- time system

6.1 signal flow graph representation of linear constant-coefficient difference equations Figure 6.8

Figure 6.9 node ： source 、 sink 、 network branch ： constant 、 z -1 、 1 、 -1 The value at each node in a graph is the sum of the outputs of all the branches entering the node.

EXAMPLE

transposed form transpose ： reverse the directions of all branches reverse the roles of the input and output The relationship between the input and output does not change.

6.2 basic structures for IIR system direct forms 1.direct I

Figure 6.14 strongpoint ： simple ； shortcoming ： more delay ； be sensitive to word length ； be inconvenient to adjust zeros and poles

double x[3],y[2]; while(!eof(in_file)) {for(k=3;k>0;k++)//M=3 x[k]=x[k-1]; x[0]=getc(in_file)-128; for(k=2;k>0;k++)/N=2 y[k]=y[k-1]; for(k=0,y[0]=0;k<=3;k++) y[0]+=b[k]*x[k]; for(k=1;k<=2;k++) y[0]+=a[k]*y[k] putc(out_file,y[0]+128); } realization of direct form in c program

2.direct II （ canonic direct form ）

Figure 6.15 strongpoint ： delay is reduced half

6.2.2 cascade forms reason of being nonuniform ： pairing manner of real zeros and poles ； order of cascade connection; pairing manner of zeros and poles 。

Figure 6.18 strongpoint ： lower sensitivity to coefficient quantization than that of direct form; search the least-error ones because of the effects of limited word length; pairing manner of real zeros and poles ； order of cascade connection; be convenient to adjust zeros and poles; time division multiplexing using a second order loop. shortcoming ： not as fast as parallel form

6.2.3 parallel forms strongpoint ： lower sensitivity to coefficient quantization than that of direct form; less error because of the effects of limited word length; be convenient to adjust poles; fast hardware realization. shortcoming ： can not adjust zeros; can not be used in the filters with high precision requirement of zero location, such as notch filter and narrowband bandstop filter.

Figure 6.20

supplement: lattice structure EXAMPLE strongpoint ： be not sensitive to limited word length

6.3 basic structures for FIR system direct forms Figure 6.31 transversal filter structure

while(!eof(in_file)) {for(k=3;k>0;k++)//M=3 x[k]=x[k-1]; x[0]=getc(in_file)-128; for(k=0,y[0]=0;k<=3;k++) y[0]+=h[k]*x[k]; putc(out_file,y[0]+128); } realization of direct form in c program

Figure cascade forms strongpoint ： be convenient to adjust zeros; time division multiplexing using a second order loop.

6.3.3 structures for linear-phase FIR system

Figure 6.34

Figure 6.35 strongpoint ： multiplication operation is reduced half

6.3.4 supplement: lattice structure EXAMPLE

6.4 effects of limited word length different infinite-precision realization structures ： the same result, different operation quantity, speed, storage space; different finite-precision realization structures ： different results, different error of frequency response, different difficulties to adjust frequency response 。 Reasons of error ： 1. coefficient quantity of filter’s ： frequency response alters ， even is instable; the more dense zeros and poles are, the more sensitive to effects of limited word length 2. round in operations High-order IIR should try to avoid using direct form; FIR （ generally, zeros distribute uniformly ） use linear-phase direct form widely.

summary 6.1 signal flow graph representation 6.2 basic structures for IIR system direct forms cascade forms parallel forms 6.3 basic structures for FIR system direct forms cascade forms structures for linear-phase FIR system

consideration ： complexity of realization; operation quantity and storage quantity ; sensitivity to effects of limited word length. requirement ： transformation between system function and flow graph representation; strongpoints and shortcomings of different structures.

exercises