《数字信号处理（英）》ppt课件.ppt

1,1)Transfer function classification,Transfer function classification Based on Magnitude CharacteristicsTransfer function classification based on Phase CharacteristicsTypes of Linear-Phase Transfer Function,Chat 7 LTI Discrete-Time System in the Transform Domain,2,7.1 Transfer function Classification Based on Magnitude Characteristics,In 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,2)Classification based on the form of the phase function,3,1)Ideal magnitude response,A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to one at these frequencies,and should have a frequency response equal to zero at all other frequencies,Definition,7.1.1 Digital Filters with Ideal Magnitude Response,4,The range of frequencies where the frequency response takes the value of one is called the pass-band,The range of frequencies where the frequency response takes the value of zero is called the stop-band,Explanation,Has a zero phase everywhere(in all frequencies),5,Diagrammatical Representation,Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:,Lowpass Highpass,Bandpass Bandstop,6,Ideal lowpass filter,a)Analytical Expression,b)Characteristics,Not absolutely summable,hence,the corresponding transfer function is not BIBO stable,Earlier in the course we derived the inverse DTFT of the frequency response of the ideal lowpass filter,(7.1),Not causal and is of doubly infinite length,7,The reason for its infinite length response is that have“brick wall”frequency responses,Resolve method,To develop stable and realizable transfer functions,the ideal frequency response specifications are relaxed by including a transition band between the passband and the stopband,This permits the magnitude response to decay slowly from its maximum value in the passband to the zero value in the stopband,Ideal lowpass filter,8,Moreover,the magnitude of response is allowed to vary by a small amount both in passband and stopband,Lowpass filter,7.1.2 Bounded real transfer function,9,1)Definition,A causal stable real-coefficient transfer function H(z)is defined as a bounded real(BR)transfer function if,2)Characteristics,Let xn and yn denote,respectively,theinput and output of a digital filter characterized by a BR transfer function H(z)with X(ej)and Y(ej)denoting their DTFTs,10,|H(ej)|1,Then the condition implies that,(7.5),Integrating the above from to,and applying Parsevals relation we get,(7.6),7.1.2 Bounded real transfer function,11,Example,Consider the causal Stable IIR transfer fuction,(7.3),where K is a real constant,Its square-magnitude function is given by,(7.4),12,13,On the other hand,for,the maximum value of is equal to at and the minimum value is equal to at,Here,the maximum value of is equal to at and the minimum value is equal to at,Hence,the maximum value can be made equal to 1 by choosing,in which case the minimum value becomes,14,Hence,is a BR function for,Plots of the magnitude function for,Example,7.1.3 Allpass transfer function,15,1)Definition,An llR transfer function A(z)with unity magnitude response for all frequencies,i.e.,is called an all pass transfer function,2)Analytical description,An M-th order causal real-coefficient all pass transfer function is of the form,(7.7),(7.8),16,If we denote the denominator polynomials of AM(z)as DM(z),then it follows that A(z)can be written as:,Note from the above that if is a pole of a real coefficient all pass transfer function,then it has a zero at,3)Zero and pole Characteristics,(7.9),(7.10),7.1.3 Allpass transfer function,17,It implies that the poles and zeros of a real-coefficient all pass function exhibit image-symmetry in the z-plane,The numerator of a real-coefficient all pass transfer function is said to be the mirror image polynomial of the denominator,and vice versa,then we have,(7.10),7.1.3 Allpass transfer function,18,4)Why is the AM(z)is the allpass transfer function,Now,the poles of a causal stable transfer function must lie inside the unit circle in the z-plane.Hence,all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle,Therefore,Hence,7.1.3 Allpass transfer function,19,5)The phase of the allpass transfer function,Figure below shows the principal value of the phase of the 3rd-order allpass function,(7.11),Note the discontinuity by the amount of in the phase,7.1.3 Allpass transfer function,20,Note:The unwrapped phase function is a continuous function of,7.1.3 Allpass transfer function,21,6)Properties,(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:,(7.20),7.1.3 Allpass transfer function,22,(3)Let denote the group delay functionof an allpass filter A(z),i.e.,The unwrapped phase function of a stable allpass function is a monotonically decreasing function of so that is everywhere positive in the range,The group delay of an M-th order stable real-coefficient allpass transfer function satisfies:,(7.21),7.1.3 Allpass transfer function,23,7)Simple Application,A simple but often used application of an allpass filter is as a delay equalizer,Delay equalizer(均衡),Implementation,Let G(z)be the transfer function of a digital filter designed to meet a prescribed magnitude response,The nonlinear phase response of G(z)can be corrected by cascading it with an allpass filter A(z)so that the overall cascade has a constant group delay in the band of interest,7.1.3 Allpass transfer function,24,Overall group delay is the given by the sum of the group delays of G(z)and A(z),Since,we have,7.1.3 Allpass transfer function,25,Left figures shows the group delay of a 4th order filter with the specifications,Right figure shows the group delay of the original filter cascaded with an 8th order allpass designed to equalize the group delay in the passband,7.1.3 Allpass transfer function,7.2 Transfer function classification based on Phase Characteristics,26,7.2.1 zero-phase Transfer-function,1)Introduction,In many applications,it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passband,One way to avoid any phase distortion is to make the frequency response of the filter real and with a zero phase characteristic,27,2)Zero-phase Transfer function,However,it is not possible to design a causal digital filter with a zero phase,zero-phase filtering can be very simply implemented by relaxing the causality requirement,One zero-phase filtering scheme is sketched below,7.2.1 zero-phase Transfer-function,28,It is easy to verify the above system with zero phase response in the frequency domain,Combining the above equations we get,Let,and denote the DTFTs of,and,respectively.We have,7.2.1 zero-phase Transfer-function,29,7.2.2 Linear-phaseTransfer-function,In the case of a causal transfer function with a nonzero phase response,the phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic in the frequency band of interest,1)Importance of the Linear-phase filter,2)Description of the Linear-phase filter,The most general type of a filter with a linear phase has a frequency response by,30,3)The Effect of the Linear-phase Filter on the Input Signal,If it is desired to pass input signal components in a certain frequency range undistorted in both magnitude and phase,then the transfer function should exhibit a unity magnitude response and a linear-phase response in the band of interest,Note also,The output yn of this filter to an inputis then given by,7.2.2 Linear-phaseTransfer-function,31,4)Diagrammatic description of the Linear-phase filter,Figure right shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband,Since the signal components in the stop band are blocked,the phase response in the stopband can be of any shape,7.2.2 Linear-phaseTransfer-function,32,5)Example,Determine the impulse response of an ideal lowpass filter with a linear phase response:,Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at,7.2.2 Linear-phaseTransfer-function,33,Noncausality:As before,the above filter is noncausal and of doubly infinite length,and hence,unrealizable,Resolve Method:By truncating the impulse response to a finite number of terms,a realizable FIR approximation to the ideal lowpass filter can be developed,Key Point:The truncated approximation may or may not exhibit linear phase,depending on the value of no chosen(to ensuring the symmetry),7.2.2 Linear-phaseTransfer-function,34,Example:If we choose no N/2 with N a positive integer,the truncated and shifted approximation,will be a length N+1 causal linear-phase FIR filter,7.2.2 Linear-phaseTransfer-function,35,Diagrammatic Description:Figure below shows the filter coefficients obtained using the function sinc for two diffrent values of N,N=12,N=13,7.2.2 Linear-phaseTransfer-function,36,Frequency Response,Because of the symmetry of the impulse response coefficients as indicated in the two figures,the frequency response of the truncated approximation can be expressed as:,where,called the zero-phase response or amplitude response,is a real function of,7.2.2 Linear-phaseTransfer-function,37,7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,1)Definition,Consider the two 1st-order transfer functions:,Pole Location:Both transfer functions have a pole inside the unit circle at the same location z=-a and are stable,Zero location:But the zero of H1(z)is inside the unit circle at z=-b,whereas,the zero of H2(z)is at z=-1/b,Situated in a mirror image symmetry,38,Zero and pole figure,Magnitude Characteristics:Both transfer functions have an identical magnitude function,H1(z),H2(z),7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,39,phase functions,Figure right shows the unwrapped phase responses of the two transfer functions for a=0.8 and b-0.5,H2(z)has an excess phase lag with respect to H1(z),7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,40,The Explanation of the Phase,The excess phase lag property of with respect to can also be explained by observing that we can write,Where is a stableallpass function,7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,41,Phase Relationship,The phase functions of and are thus related through,As the unwrapped phase function of a stable first-order allpass function is a negative function of,it follows from the above that has indeed an excess phase lag with respect to,7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,42,Decomposition of the Transfer Function into Minimum-Phase and allpass,Generalizing the above result,let be a causal stable transfer function with all zeros inside the unit circle and let H(z)be another causal stable transfer function satisfying,These two transfer functions are then related through where A(z)is a causal stable allpass function,7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,43,Phase Relationship:,H(z)has an excess phase lag with respect to Hm(z),A causal stable transfer function with all zeros inside the unit circle is called a minimum-phase transfer function,A causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function,Minimum-phase transfer function,7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,44,EXample,Consider the mixed-phase transfer function,We can rewrite H(z)as,7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions,45,7.3 Types of Linear-Phase Transfer Function,7.3.1 Frequency Response for a FIR Filter with a Linear-Phase,1)Linear Phase Requirements for Impulse Response,Even Symmetry:,(7.39),If H(z)is to have a linear-phase,its frequency response must be of the form,Let,46,Discussion of the,Where c and are constants,and,called the amplitude response,also called the zero-phase response,is a real function of.,Since,the amplitude response is then either an even function or an odd function of,i.e.,For a real e response,the magnitude response is an even function of,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,47,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,48,Replacing in the previous equation we get,(7.43),(7.42),(7.41),7.3.1 Frequency Response for a FIR Filter with a Linear-phase,49,Thus,the FIR filter with an even amplitude response will have a linear phase if it has a symmetric impulse response,Important conclusion,(7.44),As,we have,The above leads to the condition,with,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,50,Odd Symmetry,The above is satisfied if or,If is an odd function of,then from,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,Similarity Even Symmetry,we arrive at the condition for linear phase as,51,(7.48),Therefore,a FIR filter with an odd amplitude response will have linear-phase response if it has an antisymmetric impulse response,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,52,2)Four Types of Linear-Phase FIR Filter,Introduction:Since the length of the impulse response can be either even or odd,and symmetry or anti-symmetry we can define four type of Linear-phase FIR transfer functions,Declaration:,For an antisymmetric FIR filter of odd length,i.e.,N even,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,53,Diagrammatic Illustration,54,Type l:Symmetric Impulse Response with Odd Length,The transfer function H(z)is given by,Requirement:the degree N is even.,Because of symmetry,we have h0=h8,h1=h7,h2=h6,and h3=h5,Example:Assume N=8 for simplicity,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,55,The corresponding frequency response is then given by,Thus,we can write,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,56,The quantity inside the braces is a real function of and can assume positive or negative values in the range,Amplitude Function:,In 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 form,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,57,Phase Function and Group Delay:,The phase function here is given by,where is either 0 or,and hence,it is a linear function of,indicating a constant group delay of 4 samples,The group delay is given by,7.3.1 Frequency Response for a FIR Filter with a Linear-phase,58,Type 2:Symmetric Impulse Response with Even Length,Requirement:the degree N is odd.Example:Assume N=7 for simplicity,The