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Chapter7 LTI Discrete-Time Systems in the Transform Domain

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## Chapter7 LTI Discrete-Time Systems in the Transform Domain

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**Chapter7 LTI Discrete-Time Systems in the Transform Domain**Transfer Function Classification Types of Linear-Phase Transfer Functions Simple Digital Filters**Types of Transfer Functions**• The 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.**Types of Transfer Functions**• 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 |H(ei)|. (2) Classification based on the form of the phase function θ().**7.1 Transfer Function Classification Based on Magnitude**Characteristics • Digital Filters with Ideal Magnitude Responses • Bounded Real Transfer Function • Allpass Transfer Function**7.1.1 Digital Filters with Ideal Magnitude Responses**• A 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. • The range of frequencies where the frequency response takes the value of 1 is called the passband. • The range of frequencies where the frequency response takes the value of 0 is called the stopband.**Magnitude responses of the four popular types of ideal**digital filters with real impulse response coefficients are shown below: • The frequencies c, c1, and c2are called the cutoff frequencies.**An ideal filter has a magnitude response equal to 1 in the**passband and 0 in the stopband, and has a 0 phase everywhere. • The inverse DTFT of the frequency response HLP(ej) of the ideal lowpass filter is: hLP[n]=sincn/n, - <n< • We have shown that the above impulse response is not absolutely summable, and hence, the corresponding transfer function is not BIBO stable.**Also, hLP[n] is not causal and is of doubly infinite**length. • The remaining three ideal filters are also characterized by doubly infinite, noncausal impulse responses and are not absolutely summable. • Thus, the ideal filters with the ideal “brick wall” frequency responses cannot be realizedwith finite dimensional LTI filter.**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 0 value in the stopband. • Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband.**Typical magnitude response specifications**• of a lowpass filter are shown as:**Then the condition implies:**for all values of w 7.1.2 Bounded Real Transfer Functions • A causal stable real-coefficient transfer function H(z) is defined as a bounded real (BR) transfer function if: • Let x[n] and y[n] denote, respectively, the input and output of a digital filter characterized by a BR transfer function H(z) with X(ejω) and Y(ejω) denoting their DTFTs.**Integrating the above from -π to π, and applying**Parseval’s relation we get: • Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy. • It implies that a digital filter characterized by a BR transfer function can be viewed as a passive structure.**A causal stable real-coefficient transfer function H(z) with**is thus called a lossless bounded real (LBR) transfer function. If , then the output energy is equal to the input energy, and such a digital filter is therefore a lossless system. • The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity.**Bounded Real Transfer Functions**• Example: Consider the causal stable IIR transfer function: where K is a real constant. Its square-magnitude function is given by:**Thus, for α>0,**( |H(ejω)|2 )max = K2/(1- α)2 | ω=0 ( |H(ejω)|2 )min = K2/(1+α)2 | ω=π • On the other hand, for α<0, ( |H(ejω)|2 )max = K2/(1+α)2 | ω= π ( |H(ejω)|2 )min = K2/(1- α)2 | ω= 0 • Hence, is a BR function for K≤±(1-α).**Lowpass filter**Highpass filter**Where a is real, and .**Or a is complex, the H(z) should be: 7.1.3 Allpass Transfer Function • The magnitude response of allpass system satisfies: |A(ejω)|2=1, for all ω. • The H(z) of a simple 1th-order allpass system is:**An M-th order causal real-coefficient allpass transfer**function is of the form: If we denote polynomial: Then:**We shall use the notation to denote the**mirror-image polynomial of a degree-M polynomial DM(z) , i.e., • The numerator of a real-coefficient allpass transfer function is said to be the mirror-image polynomial of the denominator, and vice versa.**implies that the poles and zeros of a real-coefficient**allpass function exhibit mirror-image symmetry in the z-plane. • The expression:**To show that |AM(ejω)|=1 we observe that:**Therefore: Hence:**Figure below shows the principal value of the phase of the**3rd-order allpass function: • Note: The phase function of any arbitrary causal stable allpass function is a continuous function of ω.**Allpass Transfer Function**• Properties: • 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. • The magnitude function of a stable allpass function A(z) satisfies:**Let τg(ω) denote the group delay function of an allpass**filter A(z), i.e., The unwrapped phase function θc(ω) of a stable allpass function is a monotonically decreasing function of w so that τg(ω) is everywhere positive in the range 0 < w < p.**G(z)**A(z) Application of allpass system • Use allpass system to help design linear phase system---delay equalizer • 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.**Since , we have**• Overall group delay is the given by the sum of the group delays of G(z) and A(z). • Example: Figure below shows the group delay of a 4th order elliptic filter with the following specifications: ωp=0.3π, δp=1dB, δs=35dB**7.2 Transfer Function Classification Based on Phase**Characteristic • Zero-Phase Transfer Function • Linear-Phase Transfer Function • Minimum-Phase and Maximum-Phase Transfer Functions**7.2.1 Zero-Phase Transfer Function**• One 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. • However, it is impossible to design a causal digital filter with a zero phase.**v[n]**w[n] u[n] x[n] H(z) H(z) u[n]=v[-n], y[n]=w[-n] • Only for non-real-time processing of real-valued input signals of finite length, the zero phase condition can be met. • One zero-phase filtering scheme is sketched below: • The function filtfilt implements the above zero-phase filtering scheme.**h[n]**x[n] y[n] + Time- reversal h[n] Time- reversal • Another zero-phase filtering scheme is sketched in book P412—P7.5.**7.2.3 Minimum-Phase and Maximum-Phase Transfer Function**• Consider the two 1st-order transfer functions: Both transfer functions have a pole inside the unit circle at the same location z=-a and are stable. • 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.**H1(z)**H2(z) • Figure below shows the pole-zero plots of the two transfer functions: • Both transfer functions have an identical magnitude function.**The corresponding phase functions are:**• Figure shows the unwrapped phase responses of the two transfer functions for a = 0.8 and b =-0.5**From this figure it follows that H2(z) has an excess phase**lag with respect to H1(z). • The excess phase lag property of H2(z) with respect to H1(z) can also be explained by observing that: Where A(z) is a stable allpass function.**The phase functions of H1(z) and H2(z) are thus related**through: arg[H2(ejω)]= arg[H1(ejω)]+ arg[A(ejω)] • As the unwrapped phase function of a stable first-order allpass function is a negative function of ω, it follows from the above that H2(z) has indeed an excess phase lag with respect to H1(z).**Generalizing the above result, letHm(z) be a causal stable**transfer function with all zeros inside the unit circle and let H(z) be another causal stable transfer function satisfying |H(ejω)|= |Hm(ejω)|; • These two transfer functions are then related through H(z)=Hm(z)A(z) where A(z) is a causal stable allpass function. • The unwrapped phase functions of Hm(z) and H(z) are thus related through: arg[H(ejω)]= arg[Hm(ejω)]+ arg[A(ejω)] • 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 all zeros outside the unit circle is called a maximum-phase transfer function. • Any nonminimum-phase transfer function can be expressed as the product of a minimum-phase transfer function and a stable allpass transfer function. • A min-phase causal stable transfer function Hm(z) also has a delay that is smaller than the group delay of a nonmin-phase system which have the same magnitude response.**Example7.4:**consider the mixed-phase transfer function: • We can rewrite H(z) as**7.2.2 Linear-Phase Transfer Function**• The 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-jωD • Note: |H(ej)|=1, ()=D • The output y[n] of this filter to an input x[n]=Aejn is then given by: y[n]= Ae-jDejn =Aej(n-D)**If xa(t) and ya(t) represent the continuous-time signals**whose sampled versions, sampled at t = nT, are x[n] and y[n] given above, then the delay between xa(t) and ya(t) is precisely the group delay of amount D. • If D is aninteger, then y[n] is identical to x[n], but delayed by D samples. • If D is not an integer, y[n], being delayed by a fractional part, is not identical to x[n].**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. • Since the signal components in the stopband are blocked, the phase response in the stopband can be of any shape.**Figure right shows the frequency response if a lowpass**filter with a linear-phase characteristic in the passband**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:**^**• By truncating the impulse response to a finite number of terms, a realizable FIR approximation to the ideal lowpass filter can be developed. • The truncated approximation may or may not exhibit linear phase, depending on the value of n0 chosen. • If we choose n0= N/2 with N a positive integer, the truncated and shifted approximation:**~**^ ^ • Figure below shows the filter coefficients obtained using the function sinc for two different values of N.**7.3 Types of Linear-Phase FIR Transfer Functions**• It is nearly impossible to design a linear-phase IIR transfer function. • It is always possible to design an FIR transfer function with an exact linear-phase response. • We now develop the forms of the linear-phase FIR transfer function H(z) with real impulse response h[n].**Where c and βare constants, and , called the**amplitude function, also called the zero-phase response, is a real function of ω. 7.3.1 Linear-Phase FIR Transfer Functions • Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N: • If H(z) is to have a linear-phase, its frequency response must be of the form:**Since , the amplitude response is**then either an even or an odd function of ω, i.e., • For a real impulse response, the magnitude response |H(ejω)| is an even function of , i.e., |H(ejω)| = |H(e-jω)| • The frequency response satisfies the relation H(ejω)=H*(e-jω), or equivalently,**If is an even function, then the above relation**leads to ejβ=e-jβ implying that either β=0 or β=π. Replacing ω with –ω in the equation we get: Therefore, we have**As , we have**h[n]e-jω(c+n) = h[N-n]ejω(c+N-n) Making a change of variable l=N-n, we rewrite the equation as: The above leads to the condition: when c=-N/2,h[n]=h[N-n], 0≤n≤N

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