Consider a discrete-time lowpass filter whose impulse response h[n] is known to be real

Consider a continuous-time LTI system with frequency response \(H(j \omega)=|H(j \omega)| e^{j \sphericalangle H(j \omega)}\) and real impulse response h(t). Suppose that we apply an input \(x(t)=\cos(\omega_0t+\phi_0)\) to this system. The resulting output can be shown to be of the form \(y(t)=Ax(t-t_0)\), where A is a nonnegative real number representing an amplitude-scaling factor and \(t_0\) is a time delay. (a) Express A in terms of \(|H(j \omega_0)|\). (b) Express \(t_0\) in terms of \(\sphericalangle H\left(j \omega_{0}\right)\).

Consider a discrete-time LTI system with frequency response H(eiw) iH(eiw)jei

Consider the following frequency response for a causal and stable LTI system: . 1- jw H(jw) = --.- 1 + JW (a) Show that IH(jw )I = A, and determine the value of A. (b) Determine which of the following statements is true about T(w ), the group delay of the system. (Note: T(w) = -d( -{.H(jw ))ldw, where -{.H(jw) is expressed in a form that does not contain any discontinuities.) 1. T( w) = 0 for w > 0 2. T( w) > 0 for w > 0 3. T( w) < 0 for w > 0

Consider a linear-phase discrete-time LTI system with frequency response H(eiw) and real impulse response h[n]. The group delay function for such a system is defined as d . T(W) = - dw -{.H(elw), where -{.H(eiw) has no discontinuities. Suppose that, for this system, IH(ej"12)1 = 2, 4:H(ej0 ) = 0, and T (I) = 2 Determine the output of the system for each of the following inputs: (a) cos(n) (b) sinC; n + *)

Consider a continuous-time ideal bandpass filter whose frequency response is H( 'w) = { 1, We :::; lwl :::; 3we . 1 0, elsewhere (a) If h(t) is the impulse response of this filter, determine a function g(t) such that h(t) = ein1T:ct )g(t). (b) As we is increased, does the impulse response of the filter get more concentrated or less concentrated about the origin?

Consider a discrete-time ideal highpass filter whose frequency response is specified as 7T -We :::; lwl :::; 7T lwl < 7T- We (a) If h[n] is the impulse response of this filter, determine a function g[n] such that h[n] = ei:,n )g[n]. (b) As we is increased, does the impulse response of the filter get more concentrated or less concentrated about the origin?

A continuous-time lowpass filter has been designed with a passband frequency of 1,000 Hz, a stopband frequency of 1,200 Hz, passband ripple of 0.1, and stopband ripple of0.05. Let the impulse response of this lowpass filter be denoted by h(t). We wish to convert the filter into a bandpass filter with impulse response g(t) = 2h(t) cos(4,0007Tt). Assuming that IH(jw)l is negligible for lwl > 4,0001T, answer the following questions: (a) If the passband ripple for the bandpass filter is constrained to be 0.1, what are the two passband frequencies associated with the bandpass filter? (b) If the stopband ripple for the bandpass filter is constrained to be 0.05, what are the two stopband frequencies associated with the bandpass filter?

A causal, nonideallowpass filter is designed with frequency response H(ejw). The difference equation relating the input x[n] and output y[n] for this filter is specified as N M y[n] = L aky[n - k] + L bkx[n- k]. k=I k=O The filter also satisfies the following specifications for the magnitude of its frequency response: passband frequency = w P passband tolerance = B P stopband frequency = w s. stopband tolerance = Bs. Now consider a causal LTI system whose input and output are related by the difference equation N M y[n] = L( -l)kaky[n- k] + L( -l)kbkx[n- k]. k=I k=O Show that this filter has a passband with a tolerance of Bp, and specify the corresponding location of the passband.

Consider a continuous-time causal and stable LTI system whose input x(t) and output y(t) are related by the differential equation dy(t) dt + 5y(t) = 2x(t). What is the final value s(oo) of the step response s(t) of this filter? Also, determine the value of to for which s(t0) = s(oo) [I -:2 J.

For each second-order system whose frequency response is as follows, specify the straight -line approximation of the Bode magnitude plot: ( ) 250 (b) 0 02 jw+50 a (jw)2+50.5jw+25 ' (jw)2+0.2jw+I

A continuous-time LTI systemS with frequency response H(jw) is constructed by cascading two continuous-time LTI systems with frequency responses H1 (jw) and H2(jw), respectively. Figures P6.12(a) and P6.12(b) show the straight-line approximations of the Bode magnitude plots of H1 (jw) and H(jw ), respectively. Specify H2(jw).

The straight-line approximation of the Bode magnitude plot of a second-order continuous-time LTI systemS is shown in Figure P6.13. S may be constructed by either connecting two first-order systems sl and s2 in cascade or two first-order systems S3 and S4 in parallel. Determine which, if any, of the following statements are true or false. Justify your answers. (a) The frequency responses of S1 and S2 may be determined uniquely. (b) The frequency responses of S3 and S4 may be determined uniquely.

The straight-line approximation of the Bode magnitude plot of a causal and stable continuous-time LTI system S is shown in Figure P6.14. Specify the frequency response of a system that is the inverse of S.

For each of the following second-order differential equations for causal and stable LTI systems, determine whether the corresponding impulse response is underdamped, overdamped, or critically damped: (a) d:i~~r) + 4 d~;~n + 4y(t) = x(t) (b) sd~i~;f) +4d~~~t) +5y(t) = 7x(t) (c) d;i~~t) + 20d~;;n + y(t) = x(t) (d) 5 d"y~t) + 4 dy(f) + Sy(t) = 7 x(t) + _!_ dx(t)

A particular first-order causal and stable discrete-time LTI system has a step response whose maximum overshoot is 50% of its final value. If the final value is 1, determine a difference equation relating the input x[n] and output y[n] of this filter.

For each of the following second-order difference equations for causal and stable LTI systems, determine whether or not the step response of the system is oscillatory: (a) y[n] + y[n- 1] + ~y[n- 2] = x[n] (b) y[n] - y[n- 1] + ~y[n- 2] = x[n]

Consider the continuous-time LTI system implemented as the RC circuit shown in Figure P6.18. The voltage source x(t) is considered the input to this system. The voltage y(t) across the capacitor is considered the system output. Is it possible for the step response of the system to have an oscillatory behavior?

Consider the continuous-time LTI system implemented as the RLC circuit shown in Figure P6.19. The voltage source x(t) is considered the input to this system. The voltage y(t) across the capacitor is considered the system output. How should R, L, and C be related so that there is no oscillation in the step response?

A causal LTI filter has the frequency response H(jw) shown in Figure P6.21. For each of the input signals given below, determine the filtered output signal y(t). (a) x(t) = ejt (b) x(t) = (sinwot)u(t) (c) X(jw) = (jw)(~+ jw) (d) X(jw) = 2+ 1 jw

Shown in Figure P6.22(a) is the frequency response H(jw) of a continuous-time filter referred to as a lowpass differentiator. For each of the input signals x(t) below, determine the filtered output signal y(t). (a) x(t) = cos(21Tt + 0) (b) x(t) = cos(41Tt+O) (c) x(t) is a half-wave rectified sine wave of period, as sketched in Figure P6.22(b). IH (jw}l { sin21Tt, m:::; t:::; (m+ ~) x(t) = 0, (m + ~) :::; t :::; m for any integer m

Shown in Figure P6.23 is IH(jw)l for a lowpass filter. Determine and sketch the impulse response of the filter for each of the following phase characteristics: (a) 1:-H(jw) = 0 (b) 1:-H(jw) = wT, where Tis a constant (c) -9-.H(jw) = { ~~ 2 ' w >0 w <0

Consider a continuous-time lowpass filter whose impulse response h(t) is known to be real and whose frequency response magnitude is given as: IH(jw )I = { 1, 0, Jw I ::; 2001T otherwise (a) Determine and sketch the real-valued impulse response h(t) for this filter when the corresponding group delay function is specified as: (i) r(w) = 5 (ii) r(w) = ~ (iii) r(w) = -~ (b) If the impulse response h(t) had not been specified to be real, would knowledge of IH(jw )J and r(w) be sufficient to determine h(t) uniquely? Justify your answer.

By computing the group delay at two selected frequencies, verify that each of the following frequency responses has nonlinear phase. (a) H(jw) = Jwl+l (b) H(jw) = (Jw~l)2 (c) H(jw) = (Jw+l)l(jw+ 2)

Consider an ideal highpass filter whose frequency response is specified as H(jw) = { 1 0, Jwl >we otherwise (a) Determine the impulse response h(t) for this filter. (b) As We is increased, does h(t) get more or less concentrated about the origin? (c) Determine s(O) and s(oo), where s(t) is the step response of the filter.

The output y(t) of a causal LTI system is related to the input x(t) by the differential equation dy(t) ----;[[ + 2y(t) = x(t). (a) Determine the frequency response H(. ) = Y(jw) JW X(jw) of the system, and sketch its Bode plot. (b) Specify, as a function of frequency, the group delay associated with this system. (c) If x(t) = e-t u(t), determine Y(jw ), the Fourier transform of the output. (d) Using the technique of partial-fraction expansion, determine the output y(t) for the input x(t) in part (c). (e) Repeat parts (c) and (d), first if the input has as its Fourier transform (i) X(jw) = l+~w 2+jw' then if (ii) X(jw) = 2+jw l+jw' and finally, if (iii) X(jw) = (2+jw)l(l+jw)

(a) Sketch the Bode plots for the following frequency responses: (i) 1 + (jw/10) (ii) 1 - (jw/10) (iii) 16 (iv) 1-(jw/10) (Jw+2)4 l+jw (v) (jw/10)-1 (vi) I +(jw/10) I+Jw I+Jw (vii) 1-(jw/10) (viii) IO+Sjw+ IO(jw)2 (jw)2+(jw)+l I +(jw/10) (ix) 1 + jw + (jw)2 (x) 1- jw + (jw)2 ( ") (jw+ 10)(10jw+ I) XI [(jw/100+ I )][((jw )2 + jw +I)] (b) Determine and sketch the impulse response and the step response for the system with frequency response (iv). Do the same for the system with frequency response (vi). The system given in (iv) is often referred to as a non-minimum-phase system, while the system specified in (vi) is referred to as being a minimum phase. The corresponding impulse responses of (iv) and (vi) are referred to as a non -minimum-phase signal and a minimum-phase signal, respectively. By comparing the Bode plots of these two frequency responses, we can see that they have identical magnitudes; however, the magnitude of the phase of the system of (iv) is larger than for the system of (vi). We can also note differences in the time-domain behavior of the two systems. For example, the impulse response of the minimum-phase system has more of its energy concentrated near t = 0 than does the impulse response of the non-minimum-phase system. In addition, the step response of (iv) initially has the opposite sign from its asymptotic value as t ~ oo, while this is not the case for the system of (vi). The important concept of minimum- and non -minimum-phase systems can be extended to more general LTI systems than the simple first-order systems we have treated here, and the distinguishing characteristics of these systems can be described far more thoroughly than we have done.

An LTI system is said to have phase lead at a particular frequency w = w0 if 0. The terminology stems from the fact that if e.iwot is the input to this system, then the phase of the output will exceed, or lead, the phase of the input. Similarly, if has phase lag for all w > 0, while the system with frequency response 1 + jwT has phase lead for all w > 0. Chap.6 (a) Construct the Bode plots for the following two systems. Which has phase lead and which phase lag? Also, which one amplifies signals at certain frequencies? (i) 1 +(jw/10) (ii) 1 + IOjw I+ IOjw I +(jw/10) (b) Repeat part (a) for the following three frequency responses: (") (1 +(jw/10))2 (ii) I+ jw/10 (iii) 1 + 10jw I (1 + 10jw)3 100(jw)2+ 10jw+ I O.Ol(jw)2+0.2jw+ I

An integrator has as its frequency response H(jw) = _;_ + 7T 8(w ), )W where the impulse at w = 0 is a result of the fact that the integration of a constant input from t = -oo results in an infinite output. Thus, if we avoid inputs that are constant, or equivalently, only examine H(jw) for w > 0, we see that 20logiH(Jw)l = -20log(w), -7T the form 1 H(jw) = jw(1 + jw/10) + 7T o(w ). Sketch the Bode plot for the system for w > 0.001. (b) Sketch the Bode plot for a differentiator. (c) Do the same for systems with the following frequency responses: (i) H(jw) = I+J:::noo (ii) H(jw) = (1 +(jw)/16:(jw)211QO)

Consider the system depicted in Figure P6.32. This "compensator" box is a continuous-time LTI system. (a) Suppose that it is desired to choose the frequency response of the compensator so that the overall frequency response H(jw) of the cascade satisfies the following two conditions: 1. The log magnitude of H(jw) has a slope of -40 dB/decade beyond w = 1,000. 2. For 0 < w < 1,000, the log magnitude of H(jw) should be between -10 dB and 10 dB. Design a suitable compensator (that is, determine a frequency response for a compensator that meets the preceding requirements), and draw the Bode plot for the resulting H (j w). (b) Repeat (a) if the specifications on the log magnitude of H(jw) are as follows: 1. It should have a slope of +20 dB/decade for 0 < w < 10. 2. It should be between+ 10 and +30 dB for 10 < w < 100. 3. It should have a slope of -20 dB/decade for 100 < w < 1,000. 4. It should have a slope of -40 dB/decade for w > 1,000.

Figure P6.33 shows a system commonly used to obtain a highpass filter from a lowpass filter and vice versa. (a) Show that, if H(jw) is a lowpass filter with cutoff frequency w1P, the overall system corresponds to an ideal highpass filter. Determine the system's cutoff frequency and sketch its impulse response. (b) Show that, if H(jw) is an ideal highpass filter with cutoff frequency whP' the overall system corresponds to an ideallowpass filter, and determine the cutoff frequency of the system. (c) If the interconnection of Figure P6.33 is applied to an ideal discrete-time lowpass filter, will the resulting system be an ideal discrete-time highpass filter?

In Problem 6.33, we considered a system commonly used to obtain a highpass filter from a lowpass filter and vice versa. In this problem, we explore the system further and, in particular, consider a potential difficulty if the phase of H(jw) is not properly chosen. (a) Referring to Figure P6.33, let us assume that H(jw) is real and as shown in Figure P6.34. Then 1 - o1 < H(jw) < 1 + o1, 0 ::; w ::; w 1, -02 < H(jw) < +02, w2 < w. Determine and sketch the resulting frequency response of the overall system of Figure P6.33. Does the resulting system correspond to an approximation to a highpass filter? (b) Now let H(jw) in Figure P6.33 be of the form (P6.34-1) where H1 (jw) is identical to Figure P6.34 and O(w) is an unspecified phase characteristic. With H(jw) in this more general form, does it still correspond to an approximation to a lowpass filter? (c) Without making any assumptions about O(w ), determine and sketch the tolerance limits on the magnitude of the frequency response of the overall system of Figure P6.33. (d) If H(jw) in Figure P6.33 is an approximation to a lowpass filter with unspecified phase characteristics, will the overall system in that figure necessarily correspond to an approximation to a highpass filter?

Shown in Figure P6.35 is the frequency response H(eiw) of a discrete-time differentiator. Determine the output signal y[n] as a function of w0 if the input x[n] is x[n] = cos[won + 0].

Consider a discrete-time lowpass filter whose impulse response h[n] is known to be real and whose frequency response magnitude in the region -7r :::s w :::s 7T is given as: iwi:::; * otherwise Determine and sketch the real-valued impulse response h[n] for this filter when the corresponding group delay function is specified as: (a) T( W) = 5 (b) T( W) = ~ (C) T( W) = - ~

Consider a causal LTI system whose frequency response is given as: . . 1 - lejw H(elw) = e- JW 2 .. 1 - le- JW 2 (a) Show that iH(ejw)i is unity at all frequencies. (b) Show that 1:-H ( ejw) = - w - 2 tan- 1 ( ! sin w ) 1 - ! cosw (c) Show that the group delay for this filter is given by 3 4 T(w) = i- cosw. Sketch T(w ). (d) What is the output of this filter when the input is cos( ~n)?

Consider an ideal bandpass filter whose frequency response in the region -7T ::; w :::s 7T is specified as H(elw) = ' . { 1 0, 2f- We :::; iwi :::; otherwise 1T 2 +we Determine and sketch the impulse response h[n] for this filter when (a) We = ~ (b) We = * (c) We = ~ As we is increased, does h[n] get more or less concentrated about the origin?

Sketch the log magnitude and phase of each of the following frequency responses. (a) 1 + ~e- Jw (b) 1 + 2e- Jw (c) 1 - 2e- Jw (d) 1 + 2e-i 2w l (O l+~e-jw (e) (l + ~ r jw )3 l - ~ e- jw (g) 1+2e-;w (h) 1-2e-jw l+~e-jw 1+~e-Jw (i) 1 (l - ~ r jw )(l - ~ e-jw) (k) 1+2e-Zjw (1-~e-iw)2 (j) 1 (l- ~ r jw )(l + ~ e- jw

A particular causal LTI system is described by the difference equation j2 1 y[n]- Ty[n- 1] + 4 y[n- 2j = x[n]- x[n- 1]. (a) Find the impulse response of this system. (b) Sketch the log magnitude and the phase of the frequency response of the system.

(a) Consider two LTI systems with the following frequency responses: Show that both of these frequency responses have the same magnitude function [i.e., \H1 (e.iw)\ = \H2(e.iw)\], but the group delay of H 2(e.iw) is greater than the group delay of H 1 (e.i(u) for w > 0. (b) Determine and sketch the impulse and step responses of the two systems. (c) Show that H2(ejw) = G(e.iw)HI (eiw), where G(e.i(u) is an all-pass s_vstem [i.e., \G(e.iw)\ = 1 for all w ].

When designing filters with highpass or bandpass characteristics, it is often convenient first to design a lowpass filter with the desired passband and stopband specifications and then to transform this prototype filter to the desired highpass or bandpass filter. Such transformations are called lowpass-to-highpass or highpass-to-lowpass transformations. Designing filters in this manner is convenient because it requires us only to formulate our filter design algorithms for the class of filters with lowpass characteristics. As one example of such a procedure, consider a discrete-time lowpass filter with impulse response h1P[n] and frequency response H 1p(eiw), as sketched in Figure P6.43. Suppose the impulse response is modulated with thesequence ( -1) 11 to obtain hhp[n] = ( -1) 11 h1p[n]. (a) Determine and sketch Hhp( e.iw) in terms of H 1p( e.iw ). Show in particular that, for H 1p(e.iw) as shown in Figure P6.43, Hhp(e.iw) corresponds to a highpass filter. (b) Show that modulation of the impulse response of a discrete-time high pass filter by ( -1 ) 11 will transform it to a lowpass filter.

A discrete-time system is implemented as shown in Figure P6.44. The system S shown in the figure is an LTI system with impulse response h1p[n]. x[n] (a) Show that the overall system is time invariant. (b) If h1p[n] is a lowpass filter, what type of filter does the system of the figure implement?

Consider the following three frequency responses for causal and stable third-order LTI systems. By utilizing the properties of first- and second-order systems discussed in Section 6.6, determine whether or not the impulse response of each of the third-order systems is oscillatory. (Note: You should be able to answer this question without taking the inverse Fourier transforms of the frequency responses of the third-order systems.) H1 (e.iw) = (1 - .!.e-.iw)(l - .!.e-.iw)(l - .!.e-.iw)' 2 3 4 1 1 H2(e.iw) = ---:---------=------:---- (1 + ~e- .iw)(l - .!.e-.iw)(1 - .!.e-.iw)' - 3 4 1 H3(ejw) = -----------,--------,------- (1 - 4e-.iw)(l - ~e- jw + ~e- j2w)

Consider a causal, nonrecursive (FIR) filter whose real-valued impulse response h[n] is zero for n ~ N. (a) Assuming that N is odd, show that if h[n] is symmetric about (N- 1)/2 (i.e., if h[(N- 1)/2 + n] = h[(N- 1)/2- n]), then H(e.iw) = A(w)e-j[(N-I)!2Jw, where A(w) is a real-valued function of w. We conclude that the filter has linear phase. (b) Give an example of the impulse response h[n] of a causal, linear-phase FIR filter such that h[n] = 0 for n ~ 5 and h[n] =/= 0 for 0 ::; n ::; 4. (c) Assuming that N is even, show that if h[n] is symmetric about (N- 1)/2 (i.e., if h[(N/2) + n] = h[N/2- n- 1]), then H(e.iw) = A(w)e-j[(N-1)12lw, where A(w) is a real-valued function of w. (d) Give an example of the impulse response h[n] of a causal, linear-phase FIR filter such that h[n] = 0 for n ~ 4 and h[n] =/= 0 for 0 ::; n ::; 3.

A three-point symmetric moving average, referred to as a weighted moving average, is of the form y[n] = b{ax[n - 1] + x[n] + ax[n + 1]}. (P6.47-1) (a) Determine, as a function of a and b, the frequency response H(e.iw) of the threepoint moving average in eq. (P6.47-1). (b) Determine the scaling factor b such that H ( e.iw) has unity gain at zero frequency. (c) In many time-series analysis problems, a common choice for the coefficient a in the weighted moving average in eq. (P6.47-1) is a = 112. Determine and sketch the frequency response of the resulting filter.

Consider a four-point, moving-average, discrete-time filter for which the difference equation is y[n] = box[n] + b1 x[n- 1] + b2x[n- 2] + b3x[n- 2]. Determine and sketch the magnitude of the frequency response for each of the following cases: (a) bo = b3 = 0, b1 = b2 (b) b1 = b2 = 0, bo = b3 (c) bo = b 1 = b2 = b3 (d) bo = -hi = b2 = -b3

The time constant provides a measure of how fast a first-order system responds to inputs. The idea of measuring the speed of response of a system is also important for higher order systems, and in this problem we investigate the extension of the time constant to such systems. (a) Recall that the time constant of a first-order system with impulse response h(t) = ae -at u(t), a > 0, is 1/a, which is the amount of time from t = 0 that it takes the system step response s(t) to settle within lie of its final value [i.e., s(oo) = limr~""s(t)]. Using this same quantitative definition, find the equation that must be solved in order to determine the time constant of the causal LTI system described by the differential equation d2y(t) 11 dy(t) 10 () = 9 ( ) dt2 + dt + y t X t . (P6.49-1) (b) As can be seen from part (a), if we use the precise definition of the time constant set forth there, we obtain a simple expression for the time constant of a firstorder system, but the calculations are decidedly more complex for the system of eq. (P6.49-1). However, show that this system can be viewed as the parallel interconnection of two first-order systems. Thus, we usually think of the system of eq. (P6.49-1) as having two time constants, corresponding to the two firstorder factors. What are the two time constants for this system? (c) The discussion given in part (b) can be directly generalized to all systems with impulse responses that are linear combinations of decaying exponentials. In any system of this type, one can identify the dominant time constants of the system, which are simply the largest of the time constants. These represent the slowest parts of the system response, and consequently, they have the dominant effect on how fast the system as a whole can respond. What is the dominant time constant of the system of eq. (P6.49-1)? Substitute this time constant into the equation determined in part (a). Although the number will not satisfy the equation exactly, you should see that it nearly does, which is an indication that it is very close to the time constant defined in part (a). Thus, the approach we have outlined in part (b) and here is of value in providing insight into the speed of response of LTI systems without requiring excessive calculation. (d) One important use of the concept of dominant time constants is in the reduction of the order of LTI systems. This is of great practical significance in problems involving the analysis of complex systems having a few dominant time constants and other very small time constants. In order to reduce the complexity of the model of the system to be analyzed, one often can simplify the fast parts of the system. That is, suppose we regard a complex system as a parallel interconnection of first- and second-order systems. Suppose also that one of these subsystems, with impulse response h(t) and step response s(t), is fast-that is, that s(t) settles to its final values( oo) very quickly. Then we can approximate this subsystem by the subsystem that settles to the same final value instantaneously. That is, if s(t) is the step response to our approximation, then s(t) = s(oo)u(t). This is illustrated in Figure P6.49. Note that the impulse response of the approximate system is then h(t) = s(oo)o(t), which indicates that the approximate system is memory less. Consider again the causal LTI system described by eq. (P6.49-l) and, in particular, the representation of it as a parallel interconnection of two first -order systems, as described in part (b). Use the method just outlined to replace the faster of the two subsystems by a memory less system. What is the differential equation that then describes the resulting overall system? What is the frequency response of this system? Sketch IH(jw )I (not log IH(jw )I) and

An ideal bandpass filter is a bandpass filter that passes only a range of frequencies, without any change in amplitude or phase. As shown in Figure P6.5l(a), let the passband be w w wo - 2 :::; lwl :::; wo + 2 (a) What is the impulse response h(t) of this filter? (b) We can approximate an ideal bandpass filter by cascading a first-order lowpass and a first-order highpass filter, as shown in Figure P6.5l(b). Sketch the Bode diagrams for each of the two filters H 1 (jw) and H2(jw ). (c) Determine the Bode diagram for the overall bandpass filter in terms of your results from part (b).

In Figure P6.52(a), we show the magnitude of the frequency response for an ideal continuous-time differentiator. A nonideal differentiator would have a frequency response that is some approximation to the frequency response in the figure. (a) Consider a nonideal differentiator with frequency response G(jw) for which IG(jw )I is constrained to be within 10% of the magnitude of the frequency response of the ideal differentiator at all frequencies; that is, -O.liH(jw)l :::; [IG(jw)I-IH(jw)IJ :::; O.liH(jw)l. Sketch the region in a plot of G(jw) vs. w where IG(jw )I must be confined to meet this specification. (b) The system in Figure P6.52(b), incorporating an ideal delay ofT seconds, is sometimes used to approximate a continuous-time differentiator. ForT = 10-2 second, determine the frequency range over which the magnitude of the frequency response of the system in the figure is within 10% of that for an ideal differentiator.

In many filtering applications, it is often undesirable for the step response of a filter to overshoot its final value. In processing pictures, for example, the overshoot in the step response of a linear filter may produce flare-that is, an increase in intensityat sharp boundaries. It is possible, however, to eliminate overshoot by requiring that the impulse response of the filter be positive for all time. Show that if h(t), the impulse response of a continuous-time LTI filter, is always greater than or equal to zero, the step response of the filter is a monotonically nondecreasing function and therefore will not have overshoot.

By means of a specific filter design procedure, a nonideal continuous-time lowpass filter with frequency response H0(jw ), impulse response h0(t), and step response s0(t) has been designed. The cutoff frequency of the filter is at w = 27T X 102 rad/sec, and the step response rise time, defined as the time required for the step response to go from 10% of its final value to 90% of its final value, is Tr = 10-2 second. From this design, we can obtain a new filter with an arbitrary cutoff frequency we by the use of frequency scaling. The frequency response of the resulting filter is then of the form Hip(jw) = Ho(jaw ), where a is an appropriate scale factor. (a) Determine the scale factor a such that H1p(jw) has a cutoff frequency of We. (b) Determine the impulse response h1p(t) of the new filter in terms of We and h0 (t). (c) Determine the step response s1p(t) of the new filter in terms of we and s0(t). (d) Determine and sketch the rise time of the new filter as a function of its cutoff frequency we. This is one illustration of the trade-off between time-domain and frequency-domain characteristics. In particular, as the cutoff frequency decreases, the rise time tends to increase.

The square of the magnitude of the frequency response of a class of continuous-time lowpass filters, known as Butterworth filters, is I . 12 1 B(jw) = 1 + (w/wc)2N. Let us define the passband edge frequency w P as the frequency below which IB(jw )1 2 is greater than one-half of its value at w = 0; that is, Now let us define the stopband edge frequency Ws as the frequency above which IB(jw )1 2 is less than 10-2 of its value at w = 0; that is, The transition band is then the frequency range between w P and w5 The ratio wsfw P is referred to as the transition ratio. For fixed w P' and making reasonable approximations, deteimine and sketch the transition ratio as a function of N for the class of Butterworth filters.

In this problem, we explore some of the filtering issues involved in the commercial version of a typical system that is used in most modem cassette tape decks to reduce noise. The primary source of noise is the high-frequency hiss in the tape playback process, which, in some part, is due to the friction between the tape and the playback head. Let us assume that the noise hiss that is added to the signal upon playback has the spectrum of Figure P6.56(a) when measured in decibels, with 0 dB equal to the signal level at 100Hz. The spectrum S(j\(\omega\)) of the signal has the shape shown in Figure P6.56(b ). The system that we analyze has a filter H1 (j\(\omega\)) which conditions the signal s(t) before it is recorded. Upon playback, the hiss n(t) is added to the signal. The system is represented schematically in Figure P6.56(c). Suppose we would like our overall system to have a signal-to-noise ratio of 40 dB over the frequency range 50 Hz<\(\omega\)l\(\pi\)T <20kHz. (a) Determine the transfer characteristic of the filter H1 (j\(\omega\)). Sketch the Bode plot of H1 (j\(\omega\)). (b) If we were to listen to the signal p(t), assuming that the playback process does nothing more than add hiss to the signal, how do you think it would sound? (c) What should the Bode plot and transfer characteristic of the filter H2(j\(\omega\)) be in order for the signal s(t) to sound similar to s(t)?

Show that if h[n], the impulse response of a discrete-time LTI filter, is always greater than or equal to zero, the step response of the filter is a monotonically nondecreasing function and therefore will not have overshoot.

In the design of either analog or digital filters, we often approximate a specified magnitude characteristic without particular regard to the phase. For example, standard design techniques for lowpass and bandpass filters are typically derived from a consideration of the magnitude characteristics only. In many filtering problems, one would ideally like the phase characteristics to be zero or linear. For causal filters, it is impossible to have zero phase. However, for many digital filtering applications, it is not necessary that the unit sample response of the filter be zero for n < 0 if the processing is not to be carried out in real time. One technique commonly used in digital filtering when the data to be filtered are of finite duration and stored, for example, on a disc or magnetic tape is to process the data forward and then backward through the same filter. Let h[n] be the unit sample response of a causal filter with an arbitrary phase characteristic. Assume that h[n] is real, and denote its Fourier transform by H(ejw). Let x[n] be the data that we want to filter. The filtering operation is performed as follows: (a) Method A: Process x[n] to get s[n], as indicated in Figure P6.58(a). 507 1. Determine the overall unit sample response h 1 [n] that relates x[n] and s[n], and show that it has zero phase characteristic. 2. Determine IH1 (eiw)l and express it in terms of IH(eiw)l and are given the bandpass filter h[n] with frequency response as specified in Figure P6.58(c) and with magnitude cparacteristic that we desire, but with linear phase. To achieve zero phase, we could use either of the preceding methods, A or B. Determine and sketch IH1 (e.iw)l and IH2(e.iw)1. From these results, which method would you use to achieve the desired bandpass filtering operation? Explain why. More generally, if h[n] has the desired magnitude, but a nonlinear phase characteristic, which method is preferable to achieve a zero phase characteristic?

Let hc~[n] denote the unit sample response of a desired ideal system with frequency response Hc~(e.iw), and let h[n] denote the unit sample response for an FIR system of length Nand with frequency response H(e.iw). In this problem, we show that a rectangular window of length N samples applied to he~ [ n] will produce a unit sample response h[n] such that the mean square error is minimized. (a) The error function E(e.iw) = Hc~(e.iw)- H(e.iw) can be expressed as the power series n=-x Find the coefficients e[n] in terms of hc~[n] and h[n]. (b) Using Parsevars relation, express the mean square error E 2 in terms of the coefficients e[n]. (c) Show that for a unit sample response h[n] of length N samples, E 2 is minimized when h[n] = { hc~[n], 0, O~n~N-1 otherwise That is, simple truncation gives the best mean square approximation to a desired frequency response for a fixed value of N.

In many situations we have available an analog or digital filter module, such as a basic hardware element or computer subroutine. By using the module repetitively or by combining identical modules, it is possible to implement a new filter with improved passband or stopband characteristics. In this and the next problem, we consider two procedures for doing just that. Although the discussion is phrased in terms of discrete-time filters, much of it applies directly to continuous-time filters as well. Consider a lowpass filter with frequency response \(H\left(e^{j \omega}\right)\) for which \(\left|H\left(e^{j \omega}\right)\right|\) falls within the tolerance limits shown in Figure P6.61; that is, \(\begin{aligned}1-\delta_{1} & \leq\left|H\left(e^{j \omega}\right)\right| \leq 1+\delta_{1}, \quad 0 \leq\omega \leq \omega_{1} \\0 & \leq\left|H\left(e^{j \omega}\right)\right| \leq \delta_{2}, \omega_{2} \leq \omega \leq\pi\end{aligned}\) A new filter with frequency response \(G\left(e^{j \omega}\right)\) is formed by cascading two identical filters, both with frequency response \(H\left(e^{j \omega}\right)\)). (a) Determine the tolerance limits on \(\left|G\left(e^{j \omega}\right)\right|\). (b) Assuming that \(H\left(e^{j \omega}\right)\)) is a good approximation to a lowpass filter, so that \(\delta_{1}<<1\)1 and 8\(\delta_{2}<<1\), determine whether the passband ripple for \(G\left(e^{j \omega}\right)\) is larger or smaller than the passband ripple for \(H\left(e^{j \omega}\right)\)). Also, determine whether the stopband ripple for \(G\left(e^{j \omega}\right)\) is larger or smaller than the stopband ripple for \(H\left(e^{j \omega}\right)\)). (c) If N identical filters with frequency response \(H\left(e^{j \omega}\right)\)) are cascaded to obtain a new frequency response \(G\left(e^{j \omega}\right)\), then, again assuming that \(\delta_{1}<<1\)1 and \(\delta_{2}<<1\), determine the approximate tolerance limits on \(\left|G\left(e^{j \omega}\right)\right|\).

In Problem 6.61, we considered one method for using a basic filter module repcti tively to implement a new filter with improved characteristics. Let us now con"idet an alternative approach, proposed by J. W. Tukey in the book, Explorator." I >uru Analysis (Reading, MA: Addison-Wesley Publishing Co., Inc., 1976). The pwcedure is shown in block diagram form in Figure P6.62(a). (a) Suppose that H ( eiw) is real and has a passband ripple of :: o 1 and d stopband ripple of ::82 (i.e., H(e.iw) falls within the tolerance limits indicated in Figure P6.62(b)). The frequency response G(eiw) of the overall system in Figure P6.62(a) falls within the tolerance limits indicated in Figure P6.62(c). Determine A, B, C, and Din terms of 81 and 82. (b) If o 1 < < I and 82 < < 1, what is the approximate passband ripple and stopband ripple associated with G(eiw)? Indicate in particular whether the passband ripple for G(e.i(u) is larger or smaller than the passband ripple for H(eiw). Also, indicate whether the stopband ripple for G(eiw) is larger or smaller than the stopband ripple for H(e.iw). (c) In parts (a) and (b), we assumed that H(eiw) is real. Now consider H(e.iw) to have the more general form H(ejw) = Hl(e.iw)eje(wl, where H 1 (eiJ) is real and O(w) is an unspecified phase characteristic. If IH(e.iw)l is a reasonable approximation to an ideallowpass filter, williG(eiw)l necessarily be a reasonable approximation to an ideallowpass filter? (d) Now assume that H(efw) is an FIR linear-phase lowpass filter, so that H(eiw) = HI (efw)efMw, where H 1 ( eiw) is real and M is an integer. Show how to modify the system in Figure P6.62(a) so that the overall system will approximate a lowpass filter.

In the design of digital filters, we often choose a filter with a specified magnitude characteristic that has the shortest duration. That is, the impulse response, which is the inverse Fourier transform of the complex frequency spectrum, should be as narrow as possible. Assuming that h[n] is real, we wish to show that if the phase (}(w) associated with the frequency response H(eiw) is zero, the duration of the impulse response is minimal. Let the frequency response be expressed as and let us consider the quantity n =-ex n= -ex to be a measure of the duration of the associated impulse response h[n]. (a) Using the derivative property of the Fourier transform and Parseval's relation, express Din terms of H(eiw). (b) By expressing H(eiw) in terms of its magnitude IH(eiw)l and phase (}(w ), use your result from part (a) to show that Dis minimized when (}(w) = 0.

For a discrete-time filter to be causal and to have exactly linear phase, its impulse response must be of finite length and consequently the difference equation must be nonrecursive. To focus on the insight behind this statement, we consider a particular case, that of a linear phase characteristic for which the slope of the phase is an integer. Thus, the frequency response is assumed to be of the form (P6.64-1) where Hr(eiw) is real and even. Let h[n] denote the impulse response of the filter with frequency response H(eiw) and let hr[n] denote the impulse response of the filter with frequency response Hr(eiw). (a) By using the appropriate properties in Table 5.1, show that: 1. hr[n] = hr[-n] (i.e., hr[n] is symmetric about n = 0). 2. h[n] = hr[n - M]. (b) Using your result in part (a), show that with H(eiw) of the form shown in eq. (P6.64-1), h[n] is symmetric about n = M, that is, h[M + n] = h[M - n]. (P6.64-2) (c) According to the result in part (b), the linear phase characteristic in eq. (P6.64-1) imposes a symmetry in the impulse response. Show that if h[n] is causal and has the symmetry in eq. (P6.64-2), then h[n] = 0, n < 0 and n >2M (i.e., it must be of finite length).

For a class of discrete-time lowpass filters, known as Butterworth filters, the squared magnitude of the frequency response is given by jB(efw)l2 = 1 2N' 1 + ( tan(w/2) ) tan(wJ2) where We is the cutoff frequency (which we shall take to be 'TT'/2) and N is the order of the filter (which we shall consider to beN = 1). Thus, we have ' B(efw )j2 = 1 1 + tan2(w/2) (a) Using trigonometric identities, show that IB(efw)l2 = cos2(w/2). (b) Let B(eiw) = acos(w/2). For what complex values of a is jB(efw)l2 the same as in part (a)? (c) Show that B(efw) from part (b) is the transfer function corresponding to a difference equation of the form y[n] = ax[n] + f3x[n- y]. Determine a, f3, and 'Y.

In Figure P6.66(a) we show a discrete-time system consisting of a parallel combination of N LTI filters with impulse response hk[n], k = 0, 1, , N- 1. For any k, hk[n] is related to h0 [n] by the expression x[n] hk[n] = ei(27Tnk!N) ho[n]. (a) If h0 [n] is an ideal discrete-time lowpass filter with frequency response Ho( efw) as shown in Figure P6.66(b), sketch the Fourier transforms of h1 [n] and hN-I [n] for win the range -'TT' < w :::; +'TT'. (b) Determine the value of the cutoff frequency We in Figure P6.66(b) in terms of N (0 < we ~ 7T) such that the system of Figure P6.66(a) is an identity system; that is, y[n] = x[n] for all nand any input x[n]. (c) Suppose that h[n] is no longer restricted to be an ideallowpass filter. If h[n] denotes the impulse response of the entire system in Figure P6.66(a) with input x[n] and output y[n], then h[n] can be expressed in the form h[n] = r[n]ho[n]. Determine and sketch r[ n]. (d) From your result of part (c), determine a necessary and sufficient condition on h0 [n] to ensure that the overall system will be an identity system (i.e., such that for any input x[n], the output y[n] will be identical to x[n]). Your answer should not contain any sums.