MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems--Fall 2003 PROBLEM SET 7 Issued: October 28. 2003 Due: November 5. 2003 REMINDER: Computer Lab 2 is also due on November 7 Reading Assignments Lectures #14-15 PS#7: Chapter 7(through Section 7. 4)and Chapter 8(through Section 8.4) of o&W Lectures #16-18 PS#8: Section 7.5 and Chapters 8 and 9(through Section 9.6)of O&W Exercise for home study(not to be turned in, although we will provide solutions) O&W7.28 O&W8.23 Problems to be turned in Problem 1 A sinusoidal input signal, a(t)=cos(10t)is sampled and filtered as shown z() (t) Hw y(t) S
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 7 Issued: October 28, 2003 Due: November 5, 2003 REMINDER: Computer Lab 2 is also due on November 7. Reading Assignments: Lectures #14-15 & PS#7: Chapter 7 (through Section 7.4) and Chapter 8 (through Section 8.4) of O&W Lectures #16-18 & PS#8: Section 7.5 and Chapters 8 and 9 (through Section 9.6) of O&W Exercise for home study (not to be turned in, although we will provide solutions): O&W 7.28 O&W 8.23 Problems to be turned in: Problem 1 A sinusoidal input signal, x(t) = cos(10t) is sampled and filtered as shown below. x(t) × z(t) H(j�) y(t) s(t) 1
The frequency response of the filter lH(i)=J1,90<1-4<180 0 otherwise as shown in the figure below: H(w) ∠H(ju) 200-100 100200 (a)Suppose s(t)=28(t-kr)and T 90 Provide a labeled sketch of Z Gw), the ourier transform o (b) Find y(t), assuming the s(t)and the value of T given in part(a) (c) The sampling function s(t)is changed to the form below, with T (1) (-1)(-1)(-1)(-1) Find y(t)
� � � � � The frequency response of the filter is |H(j�)| = 1, 0, 90 < |�| < 180 otherwise �� �H(j�) = −200, as shown in the figure below: |H(j�)| �H(j�) 1 2 � 2 −� −180 −90 0 90 180 −200 −100 100 200 − � 2� (a) Suppose s(t) = �(t − kT) and T = . Provide a labeled sketch of Z(j�), the 90 k=−� Fourier transform of z(t). (b) Find y(t), assuming the s(t) and the value of T given in part (a). 2� (c) The sampling function s(t) is changed to the form below, with T = . 90 (1) (1) (1) (1) (1) t −T 2 −T −3T 2 −2T 0 T 2 T 3T 2 2T · · · · · · (−1) (−1) (−1) (−1) Find y(t). 2
Problem 2 O&W 7.30 except let the input be re(t)=8(t-T/ 2) response of an LTI, causal system with difference equatio t let hIml be the unit sample Problem 3 Answer the same questions in O&W7.31 exce yn= ryn-2+an+ran-1 Problem 4 o&w8.22 Problem 5 o&W8.49 Problem 6 The transmission system depicted below is intended to allow a signal a(t)to be transmitted through a communication channel that also carries other signals represented Channel t) y() LPG) z(t) Both x(t)and a(t)are bandlimited, and their Fourier transforms X (u)and Z(w)are real as sketched below. Notice that the bandwidth w2 of Z(w)is much greater than the band- width wx of X(w) B
� � � � � �� � � � �� � � � � � � � �� � � � � � � � � � � � � � � � � � �� � Problem 2 O&W 7.30 except let the input be xc(t) = �(t − T/2). Problem 3 Answer the same questions in O&W 7.31 except let h[n] be the unit sample response of an LTI, causal system with difference equation 3 1 y[n] = y[n − 2] + x[n] + x[n − 1]. 4 4 Problem 4 O&W 8.22 Problem 5 O&W 8.49 Problem 6 The transmission system depicted below is intended to allow a signal x(t) to be transmitted through a communication channel that also carries other signals represented by z(t). Channel x(t)� �p(t) � q(t) r(t) +� � � � HLP (j�) � y(t) HLP (jw) × × C z(t) −�f 0 �f cos(�ct) Both x(t) and z(t) are bandlimited, and their Fourier transforms X(j�) and Z(j�) are real, as sketched below. Notice that the bandwidth �z of Z(j�) is much greater than the bandwidth �x of X(j�). X(j�) Z(j�) A B � �� �� � −�x �x −�z �z 3
(a) We wish to determine parameters for the transmission system so that the output y(t) is equal to the input (t). Determine the range of wc for which y(t) can be made equal to a(t). Explain (b) Given a value of wc in the range specified in part(a), determine the range of values of f and the value of C for which y(t)= (t). Your expression may contain wc and/or arameters of the Fourier transforms X(w)and Z(w). Briefly explain your reasoning (c) Consider next what would happen if the channel also had appreciable delay, as depicted below. Assuming that the parameters are chosen as in parts(a)and(b), find an expression for the systems frequency response, defined as Channel 4 p(t) Delay H z(t) Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for
� � � � � �� (a) We wish to determine parameters for the transmission system so that the output y(t) is equal to the input x(t). Determine the range of �c for which y(t) can be made equal to x(t). Explain. � (b) Given a value of �c in the range specified in part (a), determine the range of values of f and the value of C for which y(t) = x(t). Your expression may contain �c and/or parameters of the Fourier transforms X(j�) and Z(j�). Briefly explain your reasoning. (c) Consider next what would happen if the channel also had appreciable delay, as depicted below. Assuming that the parameters are chosen as in parts (a) and (b), find an expression for the system’s frequency response, defined as Y (j�) H(j�) = X(j�) Channel T Delay by x(t)� �p(t) � q(t) r(t) +�� � � � HLP (j�) � y(t) HLP (jw) × × C z(t) −�f 0 �f cos(�ct) Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for tests. 4