Chapter7 LTI Discrete-Time Systems in the Transform Domain Transfer Function ClassificationTypes of Linear-Phase Transfer FunctionsSimple Digital Filters 1 Types of Transfer FunctionsnThe time-domain classification of a digital transfer function based on the length of its impulse response sequence:- Finite impulse response (FIR) transfer function.- Infinite impulse response (IIR) transfer function.2 Types of Transfer FunctionsnIn the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications:(1) Classification based on the shape of the magnitude function |H(ei )|. (2) Classification based on the form of the phase function θ( ).3 7.1 Transfer Function Classification Based on Magnitude CharacteristicsnDigital Filters with Ideal Magnitude ResponsesnBounded Real Transfer FunctionnAllpass Transfer Function4 7.1.1 Digital Filters with Ideal Magnitude ResponsesnA digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to 1 at these frequencies, and should have a frequency response equal to 0 at all other frequencies.5 Digital Filters with Ideal Magnitude ResponsesnThe range of frequencies where the frequency response takes the value of 1 is called the passband.nThe range of frequencies where the frequency response takes the value of 0 is called the stopband.6 Digital Filters with Ideal Magnitude ResponsesnMagnitude responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:7 Digital Filters with Ideal Magnitude ResponsesnThe frequencies c, c1, and c2 are called the cutoff frequencies.nAn ideal filter has a magnitude response equal to 1 in the passband and 0 in the stopband, and has a 0 phase everywhere.8 Digital Filters with Ideal Magnitude ResponsesnEarlier in the course we derived the inverse DTFT of the frequency response HLP(ej )of the ideal lowpass filter: hLP[n]=sin cn/n , - 0, ( |H(ejω)|2 )max = K2/(1- α)2 | ω=0 ( |H(ejω)|2 ) min = K2/(1+ α)2 | ω=πnOn the other hand, for α<0, (2αcosω )max = -2α | ω= π (2αcosω )min = 2α | ω=0Here, ( |H(ejω)|2 )max = K2/(1+ α)2 | ω= π ( |H(ejω)|2 )min = K2/(1- α)2 | ω= 018 Bounded Real Transfer FunctionsnHence,is a BR function for K≤±(1-α),Plots of the magnitude function for α=±0.5 with values of K chosen to make H(z) a BR function are shown on the next page.19 Bounded Real Transfer FunctionsLowpass filterHighpass filter20 7.1.3 Allpass Transfer FunctionnThe magnitude response of allpass system satisfies: |A(ejω)|2=1,for all ω.nThe H(z) of a simple 1th-order allpass system is:Where a is real, and .Or a is complex ,the H(z) should be:21 Allpass Transfer Functionone real pole one complex pole 22 Allpass Transfer FunctionnTwo order allpass transfer function ploes：：zeros：：23 Allpass Transfer FunctionnGeneralize, the Mth-order allpass system is:If we denote polynomial:So:24 Allpass Transfer FunctionnThe numerator of a real-coefficient allpass transfer function is said to be the mirror-image polynomial of the denominator, and vice versa.• We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial DM(z) , i.e.,25 Allpass Transfer FunctionnThe expressionimplies that the poles and zeros of a real-coefficient allpass function exhibit mirror-image symmetry in the z-plane.26 Allpass Transfer FunctionnTo show that |AM(ejω)|=1 we observe that:•Therefore:Hence:27 Allpass Transfer FunctionnProperties:(1)A causal stable real-coefficient allpass transfer function is a lossless bounded real (LBR) function or, equivalently, a causal stable allpass filter is a lossless structure.(2)The magnitude function of a stable allpass function A(z) satisfies:28 Allpass Transfer Function(3) Let τ τg g( (ω ω) ) denote the group delay function of an allpass filter A(z), i.e.,The unwrapped phase function θ θc c( (ω ω) ) of a stable allpass function is a monotonically decreasing function of so that τ τg g( (ω ω) ) is everywhere positive in the range 0 < < . .29 Application of allpass systemnAny causal stable system can be denoted as: H(z)=Hmin(z)A(z)Where Hmin(z) is a minimum phase-delay system.nUse allpass system to help design stable filters.nUse allpass system to help design linear phase system.nA simple example.(P361,Fig7.7)30 7.2 Transfer Function Classification Based on Phase Characteristic1、、The phase delay will cause the change of signal waveform时间时间 tAmp原始信号原始信号时间时间 t幅幅度度相移相移90o时间时间 t幅幅度度相移相移 180o31 2、、The nonlinearity of system phase delay will cause the signal distortionf1 f2f时时延延f1 f2f时时延延f1 f2f ( )f1 f2f ( )Time delay of signal is depended on systemphase characteristic32 3、、If we ignore the phase information, then输入波形输入波形DFT变换变换忽略相忽略相位信息位信息IDFT变换变换输出波形输出波形33 nlinear phase requirement：：4、、The linear phase FIR filter design ---- group delay34 7.2 Transfer Function Classification Based on Phase Characteristic nZero-Phase Transfer FunctionnLinear-Phase Transfer FunctionnMinimum-Phase and Maximum-Phase Transfer Functions35 7.2.1 Zero-Phase Transfer FunctionnOne way to avoid any phase distortions is to make the frequency response of the filter real and nonnegative,to design the filter with a zero phase characteristic.n But for a causal digital filter it is impossible.36 Zero-Phase Transfer FunctionnOnly for non-real-time processing of real-valued input signals of finite length, the zero phase condition can be met.Let H(z) be a real-coefficient rational z-transform with no poles on the unit cycle, then F(z)=H(z)H(z-1) has a zero phase on the unit cycle.37 Zero-Phase Transfer FunctionnPlease look at book P362.x[n]v[n]u[n]w[n]H(z)H(z)u[n]=v[-n], y[n]=w[-n]The function filtfilt implements the above zero-phase filtering scheme.n Please look at book P412—P7.5.38 7.2.2 Linear-Phase Transfer FunctionnThe phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic, that is:H(ejω)=e-jωDnHow to perform the linear-phase filter?y[n]=x[n-D]Y(ejω)= e-jωDX(ejω)H(ejω)= Y(ejω)/ X(ejω)= e-jωD39 Linear-Phase Transfer FunctionnExample - Determine the impulse response of an ideal lowpass filter with a linear phase response:n Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at:40 Linear-Phase Transfer FunctionnBy truncating the impulse response to a finite number of terms, a realizable FIR approximation to the ideal lowpass filter can be developed.nThe truncated approximation may or may not exhibit linear phase, depending on the value of n0 chosen.nIf we choose n0= N/2 with N a positive integer, the truncated and shifted approximation:^41 Linear-Phase Transfer FunctionnFigure below shows the filter coefficients obtained using the function sinc for two different values of N.42 7.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionnBased on the expression:The phase function is:Where, ξk andλk are zeros and poles, respectively.43 Minimum-Phase and Maximum-Phase Transfer FunctionnWe define the expression (ejω- ξk )and (ejω - λk ) as zero vectors and pole vectors.nWhen ξk and λk are inside the unit circle,and ωchange from 0 to 2π, the change of phase of the zero (pole) vectors are 2π. nWhen ξk and λk are outside the unit circle,and ωchange from 0 to 2π, the change of phase of the zero (pole) vectors are 0. 44 Minimum-Phase and Maximum-Phase Transfer FunctionnSo, only zeros or poles inside the unit circle can affect phase function of H(ejω).nFor causal stable system, we can deduce:So, it’s the phase delay (lag) system.When all zeros are all inside the unit circle, we get:It’s the minimum phase delay system.45 Minimum-Phase and Maximum-Phase Transfer FunctionWhen all zeros are all outside the unit circle, we get:Where M is the number of zeros.And it’s the maximum phase delay system.nA transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function.nExample7.4(P367).46 7.3 Types of Linear-Phase FIR Transfer FunctionsnIt is nearly impossible to design a linear-phase IIR transfer function.nIt is always possible to design an FIR transfer function with an exact linear-phase response.nWe now develop the forms of the linear-phase FIR transfer function H(z) with real impulse response h[n].47 Linear-Phase FIR Transfer FunctionsnIf H(z) is to have a linear-phase, its frequency response must be of the formWhere c and βare constants, and , called the amplitude response, also called the zero-phase response, is a real function of ω ω .•Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N:48 Linear-Phase FIR Transfer FunctionsnFor a real impulse response, the magnitude response |H(ejω)| is an even function of , i.e., |H(ejω)| = |H(e-jω)| • Since , the amplitude response is then either an even function or an odd function of ω ω, i.e.49 Linear-Phase FIR Transfer FunctionsnThe frequency response satisfies the relation H(ejω)=H*(e-jω), or equivalently, the relationIf is an even function, then the above relation leads to ejβ=e-jβ implying that either β=0 or β β= =π π50 Linear-Phase FIR Transfer FunctionsnFrom We haveSubstituting the value of β in the above we get51 Linear-Phase FIR Transfer FunctionsnReplacing ω with –ω in the previous equation we getMaking a change of variable l=N-n, we rewrite the above equation as52 Linear-Phase FIR Transfer FunctionsAs , we have h[n]e-jω(c+n)= h[N-n]ejω(c+N-n)The above leads to the condition when c=-N/2 h[n]=h[N-n], 0≤n≤NThus, the FIR filter with an even amplitude response will have a linear phase if it has a symmetric impulse response.53 Linear-Phase FIR Transfer FunctionsIf is an odd function of ω ω, then fromWe get ejβ= -e-jβ as The above is satisfied if β=π π/ /2 or β=- π/2 ，，ThenReduces to54 Linear-Phase FIR Transfer FunctionsnThe last equation can be rewritten as:As , from the above we get55 Linear-Phase FIR Transfer FunctionsnMaking a change of variable l=N-n, we rewrite the last equation as:Equating the above withWe arrive at the condition for linear phase as:56 Linear-Phase FIR Transfer Functions h[n]=-h[N-n], 0 n Nwith c=-N/2Therefore a FIR filter with an odd amplitude response will have linear-phase response if it has an antisymmetric impulse response.57 Linear-Phase FIR Transfer FunctionsnSince the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functionsnFor an antisymmetric FIR filter of odd length, i.e., N even h[N/2] = 0nWe examine next the each of the 4 cases58 Linear-Phase FIR Transfer FunctionsType 1: N = 8Type 2: N = 7Type 3: N = 8Type 4: N = 759 Linear-Phase FIR Transfer FunctionsType 1: Symmetric Impulse Response with Odd LengthnIn this case, the degree N is evennAssume N = 8 for simplicitynThe transfer function H(z) is given by60 Linear-Phase FIR Transfer FunctionsnBecause of symmetry, we have h[0]=h[8], h[1] = h[7], h[2] = h[6], and h[3] = h[5]nThus, we can write61 Linear-Phase FIR Transfer FunctionsnThe corresponding frequency response is then given by• The quantity inside the braces is a real function of , and can assume positive or negative values in the range 0 | |62 Linear-Phase FIR Transfer Functionswhere b is either 0 or p, and hence, it is a linear function of w in the generalized sensenThe group delay is given by indicating a constant group delay of 4 samples• The phase function here is given by63 Linear-Phase FIR Transfer FunctionsnIn the general case for Type 1 FIR filters, the frequency response is of the form~~where the amplitude response , also called the zero-phase response, is of the form64 Linear-Phase FIR Transfer FunctionsType 2: Symmetric Impulse Response with even LengthType 3: Antisymmetric Impulse Response with odd LengthType 4: Antisymmetric Impulse Response with even LengthP371-372 about these FIR transfer functions.65 Four Types of Linear Phase Filter 66 Four Types of Linear Phase Filter 67 Linear-Phase FIR Transfer Functions which is seen to be a slightly modified version of a length-7 moving-average FIR filternThe above transfer function has a symmetric impulse response and therefore a linear phase responseExample - Consider68 Linear-Phase FIR Transfer FunctionsnA plot of the magnitude response of along with that of the 7-point moving-average filter is shown below00.20.40.60.8100.20.40.60.81w/pMagnitudemodified filtermoving-average69 Linear-Phase FIR Transfer FunctionsnNote the improved magnitude response obtained by simply changing the first and the last impulse response coefficients of a moving-average (MA) filternIt can be shown that we can express which is seen to be a cascade of a 2-point MA filter with a 6-point MA filter• Thus, H0(z) has a double zero at z=-1, i.e., (w = p)70 7.3.1 Zero Locations of Linear-Phase FIR Transfer FunctionsnThe zeros of the real-coefficient h[n] is in mirror-image pairs.nMoreover, for a FIR filter with a real impulse response, the zeros occur in complex conjugate pairs. nParticularly, zeros position for different linear-phase types:71 Zero Locations of Linear-Phase FIR Transfer Functions(1)Type 1 :Either an even number or no zeros at z=1& z=-1.(2)Type 2 :Either an even number or no zeros at z=1,and an odd number of zeros at z=-1.(3)Type 3 :An odd number of zeros at z=1& z=-1.(4)Type 4 :An odd number of zeros at z=1, and either an even number or no zeros at z=-1.72 7.4 Simple Digital Filters nSimple FIR Digital FiltersLowpass; HighpassnSimple IIR Digital FiltersLowpass; Highpass; Bandpass; BandstopnComb FiltersFIR Comb Filters; IIR Comb Filters73 HomeworknRead the textbook from p.351 to 378nProblems 7.4, 7.5, 7.10, 7. 21, 7.28, 7.41,7.43,7.60, 7.69M7.3, M7.5, M7.674 。
