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1、离散时间信号处理DSP Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope。有生命必有希望。有生命必有希望6.0 IntroductionSeveral ways to describe discrete-time systems:Impulse responses in time domain.Difference equations in time domain.The z transforms in complex frequency domain(transfer functions).Fo
2、urier transforms in frequency domain(Frequency response).H(ej)=|H(ej)|ej()26.1 Description of the Digital Filter StructuresIts difference equations in time domain is将输入加以延时，组成 M 节的延时网络，把每个延时抽头后加权，然后把结果相加。将输出加以延时，组成 N 节的延时网络，把每个延时抽头后加权，然后把结果相加。因此，网络结构表示一定的运算结构，不同结构所需要因此，网络结构表示一定的运算结构，不同结构所需要的存储单元以及运算
3、次数不同，前者影响结构复杂性，的存储单元以及运算次数不同，前者影响结构复杂性，后者影响运算速度。后者影响运算速度。36.1 Description of the Digital Filter StructuresThree basic elements to implement digital filters:DelayMultiplierAdderBlock diagram（方框图）representation of three basic elements.z1X(z)x(n)x(n 1)z1X(z)x(n)X(z)kk X(z)k x(n)x1(n)X1(z)x2(n)X2(z)x1(n
4、)+x2(n)X1(z)+X2(z)46.1 Description of the Digital Filter StructuresSignal flowgraph（信号流图）representation of three basic elements.X(z)x(n)x(n 1)z1X(z)z1x1(n)X1(z)x2(n)X2(z)x1(n)+x2(n)X1(z)+X2(z)X(z)x(n)k x(n)k X(z)k56.1 Description of the Digital Filter StructuresTwo classes of digital filters:Finite-
5、duration impulse response filters or nonrecursive filters.Its transfer functions are of the polynomial form.Infinite-duration impulse response filters or recursive filters.Its transfer functions are of the rational polynomial form.66.3 Basic structures for IIR digital filters6.3.1 Direct form IThe t
6、ransfer function of a recursive filter is given byAnd the difference equations in time domain isIn general,M N.76.3.1 Direct forms Ix(n)y(n)z1z1b0b1b2z1bMbM1z1z1z1a1a2aN1aNy(n1)y(n2)y(nN)x(n1)x(n2)x(nM)86.3.1 Direct forms Ix(n)y(n)b0b1b2bMbM1z1z1z1a1a2aN1aNDirect forms I structure for IIR digital fi
7、lters96.3.2 Direct forms IIx(n)y(n)z1z1b0b1b2z1bMbM1z1z1z1a1a2aN1aN106.3.2 Direct forms IIx(n)y(n)b0b1b2bMbM1z1z1z1a1a2aN1aNDirect forms II structure for IIR digital filters11Comparison of the two typesx(n)y(n)b0b1b2bMbM1z1z1z1a1a2aN1aNDirect forms Ix(n)y(n)b0b1b2bMbM1z1z1z1a1a2aN1aNDirect forms II1
8、2Example 1Compute H(z)from the following signal flowgraph.Solution:x(n)y(n)1/4z1z11/4-3/82136.3.3 Cascade formWriting the numerator and denominator polynomials of H(z)as products of second-order factors,respectively,we have thatx(n)y(n)11z1z1m11m21211mz1z1m1mm2m2mH0146.3.4 Parallel formH(z)can also
9、be expressed as an addition of second-order partial-fractions,such thatx(n)y(n)11z1z1m11m21z1z1m1mm2m0m011m15Example 2Figure the signal flowgraph of the following system by the direct form(type I and II),cascade form and parallel form.Solution:x(n)y(n)1/3z1z13/4-1/8Type IIx(n)y(n)3/4-1/8z1z11/3Type
10、I16Example 2x(n)y(n)1/3z11/4z11/2Cascade form17Example 2x(n)y(n)z11/4z11/210/3-7/3Parallel form18Example 3Determine the transfer function of the system below:x(n)y(n)z11/3z11/5-15/2-3196.5 Basic structures for FIR digital filtersThe difference equation of FIR filters 206.5.1 Direct formx(n)y(n)z1z1z
11、1h(0)h(1)h(2)h(M1)h(M)216.5.1 Direct formTransposed direct form（直接型结构的转置）x(n)y(n)z1z1z1h(0)h(1)h(2)h(M1)h(M)22Example 4Compute the transfer function given by the signal flowgraph and the direct form of H(z).x(n)y(n)h(0)h(1)h(2)h(3)h(4)h(5)h(6)h(7)h(8)23Example 4x(n)y(n)z1z1z1h(0)h(1)h(2)h(3)h(4)h(5)
12、h(6)h(7)h(8)z1z1z1z1z1246.5.2 Cascade formWriting H(z)as a product of second-order factors,we get that x(n)y(n)z1z1011121z1z1021222z1z12N1N0N256.5.3 Linear-phase forms（线性相位型）An important subclass of FIR digital filters is the one that includes linear-phase filters,that isand the frequency response h
13、as the following form266.5.3 Linear-phase forms where b(n)is the inverse Fourier transform of B(),andSince B()is real,So 276.5.3 Linear-phase formsIn the common case where all the filter coefficients are real,soIf h(n)is causal,that is h(n)=0,for n 2.So,This equation shows that the h(n)of a linear-p
14、hase filter is symmetric or antisymmetric about M/2.286.5.3 Linear-phase formsnh(n)2103 4 5 6 7 8nh(n)2103 4 5 6 7 8 9nh(n)2103 45 6 7 8nh(n)2103 45 6 7 8 9symmetricantisymmetricM evenM oddType IType IIType IIIType IV296.5.3 Linear-phase forms:type I306.5.3 Linear-phase forms:type Ix(n)y(n)z1z1z1h(0
15、)h(1)h(2)h(M/2)z1z1z1h(M/21)316.5.3 Linear-phase forms:type II326.5.3 Linear-phase forms:type IIx(n)y(n)z1z1z1h(0)h(1)h(2)z1z1z1z1336.5.3 Linear-phase forms:type III346.5.3 Linear-phase forms:type IIIx(n)y(n)z1z1z1h(0)h(1)h(2)z1z1z1h(M/21)1111356.5.3 Linear-phase forms:type IV366.5.3 Linear-phase fo
16、rms:type IVx(n)y(n)z1z1z1h(0)h(1)h(2)z1z1z1z11111137Example 5Draw the signal flow-gragh of the direct form and linear-phase form for the FIR system.Solution:direct formx(n)y(n)z1z1z11-23-21z1z13-4z138Example 5Linear-phase form x(n)y(n)z1z1z11-23-4z1z1z1396.5.3 Linear-phase formsClearly,the linear-ph
17、ase form structure requires about 50%fewer multiplications than that of the direct forms.40Digital network analysisThe analysis of digital networks is realized through the signal flow graph representation.A digital network consists of three devices:delays,multipliers and adders.A signal flow graph i
18、s a network composed of directed branches and nodes.A branch delays a signal or multiplies it by a coefficient.The output value of each node is determined by the sum of all other nodes entering the node through branches.41Example 6x(n)y(n)a43a21z1z1a23a4142Example 7Determine the transfer function and the impulse response of the system below:Solution:x(n)y(n)2z1-1/2243Exercises1)Draw the signal flow-gragh of the direct form and linear-phase form for the FIR system.44Exercises2)用直接 I 型和直接 II 型结构实现以下系统函数：45