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MIT 2 161 - Convolution and Fourier Transforms

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.h(t)t1.01.00-1.00.750-1.5f(t)t(a) Impulsive waveform(b) Impulse responseh(t)t1.01.00-1.00.750-1.5f(t)t(a) Rectangular input function(b) Impulse responseMASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Problem Set 1: Convolution and Fourier Transforms Assigned: Sept. 9, 2008 Due: Sept. 18, 2008 Problem 1: (a) Plot the result of convolving a one-dimensional function consisting of three impulses with a triangular function as shown below: (b) Plot the result of convolving a one-dimensional function consisting of a rectangular function with the same triangular function used in part(a), as shown below: (c) Plot the result of convolving a pair of identical even “top-hat” (pulse) functions: f (t) = 1 for t < T/2 and f(t) = 0 otherwise. Use your result show that the Fourier transform | | 2of a triangular pulse (such as used in parts (a) and (b)) is of the form (sin(x)/x) . Problem 2: Show that when two gaussian functions 2 f1(x) = e−axf2(x) = e−bx2 are convolved, the result is another gaussian function. Hint: Complete the square in the exponent, change the variable of integration, and recognize that 2� ∞ e−xdx = √π −∞0.50-0.5f(t)tProblem 3: Find the Fourier Transform of three equally spaced impulses as shown below: Plot the result (real and imaginary parts). Problem 4: (The following is taken from the Signal Processing PhD Quals written exam for January 2007. Note: this is not the complete exam.) Assume we have a signal x(t) with a Fourier transform X(jΩ) given by X(jΩ) = ⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ 0 Ω < −2W X0/2 −2W ≤ Ω < 0 0 ≤ Ω < WX0 0 Ω ≥ W where X0 is some real valued number, W is a real valued positive number, and Ω is specified in units of radians/second. (a) What is the value of x(t) at t = 0? (b) For an arbitrary t, what is the relationship between x(t) and x(−t)? (c) What is the value of � ∞x(t)dt?−∞ 2(d) What is the value of � ∞x(t) dt?−∞ | | Problem 5: An impulse δ(t) is passed through an ideal low-pass filter with frequency response H(jΩ) = 1 for Ω < Ωc and H(jΩ) = 0 otherwise. Find and sketch y(t), the| |output of the filter. Is this a causal filter? Problem 6: After measurement and curve-fitting it is determined that a causal signal processing filter has an impulse response h(t) = 5e−3t for t > 0.. What is the filter’s (a) transfer function, and (b) its frequency response function? Determine the -6 dB cut-off frequency, that is the frequency at which the output amplitude is one half of the low frequency


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