Final Exam Solutions June 14, 2006

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1 Name or 6-Digit Code: PSU Student ID Number: Final Exam Solutions June 14, 2006 ECE 223: Signals & Systems II Dr. McNames Keep your exam flat during the entire exam. If you have to leave the exam temporarily, close the exam and leave it face down while you are out of the room. Turn off cell phones. Do not open the exam until instructed to do so. Do not use separate scratch paper. If you need more space, use the backs of the exam pages and write a note directing my attention to these pages. You will have 100 minutes to complete the exam. If you have extra time, double check your answers. Include units with each of your answers. Show all of your work for full credit. Problem 1: / 18 Problem 2: / 14 Problem 3: / 10 Problem 4: / 8 Total: / 50

2 Final Exam Solutions ECE 223 Spring of 6 1. Key Concepts (18 pts) The following abbreviations are used throughout this exam: FT = Fourier Transform, FS = Fourier Series, DT = Discrete-Time, and CT = Continuous-Time. Unless specified otherwise, you may assume the signal is real-valued for all of these questions. a. (4 pts) What are the following components of the signal x[n] = 17 exp(-j1.7πn)? Angle Imaginary part Fundamental frequency Fundamental period angle{x[n]} = -1.7πn Im{x[n]} = -17 sin(1.7πn) Ω = -1.7π N = 20 samples b. (1 pt) What types of signals can be represented as sums of sinusoids? Energy Power Periodic Nonperiodic c. (1 pt) Which types of signals can be represented as finite amplitude integrals of complex sinusoids? Energy Power Periodic Nonperiodic d. (2 pts) If a sinusoidal signal x(t) = A cos(ωt + θ) is applied to an LTI system, how is the output of the system related to the input signal and the system s impulse response h(t)? H(jω) = CTFT{h(t)}, y(t) = A H(jω) cos(ωt + θ + angle{h(jω)}) e. (2 pts) If a sinusoidal signal x[n] = A cos(ωn + θ) is applied to an LTI system, how is the output of the system related to the input signal and the system s impulse response h[n]? H(e jω ) = DTFT{ H(e jω )}, y[n] = A H(e jω ) cos(ωn + θ + angle{ H(e jω )}) f. (1 pt) Why is Parseval s theorem important? It permits us to interpret the squared magnitude of a signal s Fourier transform as an energy spectral density. This permits us to compute what fraction of the signal power is within a specified frequency range, which is useful for making design decisions. g. (1 pt) What is the difference between the DFT and the FFT? The FFT is a fast algorithm to calculate the discrete Fourier transform (DFT). The DFT is defined for all DT signals, but the FFT can only be applied to signals that have a length that is an integer power of 2 (e.g., N = 512, 1024, 2048 samples). h. (1 pt) Which of the following transforms can be calculated with a computer exactly, ignoring finite precision effects? DTFT DTFS CTFT CTFS i. (1 pt) Which of the following transforms can be estimated with the FFT? DTFT DTFS CTFT CTFS

3 Final Exam Solutions ECE 223 Spring of 6 1. Key Concepts Continued (18 pts) j. (1 pt) What is the purpose of zero padding? Zero padding serves two purposes. First, it permits the FFT to be applied to signals that do not have a length that is an integer power of 2. Second, it decreases the frequency domain sampling interval which, in turn, decreases the piecewise linear interpolation error. k. (1 pt) What is the consequence of windowing a signal when estimating the CTFT? Windowing a signal is equivalent to convolving the signal s CTFT with the window s CTFT. This results in a blurring or smoothing of the estimated CTFT of the original signal. l. (1 pt) Why is windowing almost always applied in the estimation of the CTFT? The CTFT is defined over an infinite duration, but in practice we can only observe and measure a signal for a finite duration. This process of assuming the signal is zero outside of the sample interval is called windowing. m. (1 pt) What is Gibb s phenomenon? Gibb s phenomenon describes the growing oscillations that occur at discontinuities when a sum or integral of sinusoids is used to estimate a signal. As higher frequency components are included, the duration of the oscillations decreases, but the maximum error does not.

4 Final Exam Solutions ECE 223 Spring of 6 2. DT Systems (14 pts) The frequency response of a discrete-time system is shown below. The phase of the response at Ω = 0 is Magnitude pi 0.1 pi 0.2 pi 0.3 pi 0.4 pi 0.5 pi 0.6 pi 0.7 pi 0.8 pi 0.9 pi 1.0 pi Phase (deg) pi 0.1 pi 0.2 pi 0.3 pi 0.4 pi 0.5 pi 0.6 pi 0.7 pi 0.8 pi 0.9 pi 1.0 pi Frequency (rad/sec) a. (1 pt) What type of filter is this? Lowpass Highpass Bandpass Bandstop None b. (1 pt) What type of filter is this? Elliptic Chebyshev Type I Chebyshev Type II Butterworth c. (1 pt) Suppose the minimum gain in the passband is What are the approximate passband frequencies? Include units with your answer. 0.19π < ω < 0.71π d. (1 pt) Suppose the maximum gain in the stopband is What are the approximate stopband frequencies? Include units with your answer. 0 < ω < 0.13π and 0.80π < ω < π e. (3 pts) Evaluate the frequency response at the following frequencies. H(e j2π ) = 0 angle{h(e j2π )} = 0 (given) H(e j33.1π ) = 0 angle{h(e j33.1π )} = -90 (Ω = -0.9π) H(e -j0.4π ) = 2 angle{h(e -j0.4π )} = -25 (Ω = -0.4π) f. (7 pts) Suppose the following input signal is applied to this system. What is the output of the system? Hint: use the symmetry and periodic properties of the DTFT. x[n] = sin(0.3π n + 12 ) cos(4.8π n ) + 80 cos(-0.4π n ) y[n] = sin(0.3π n ) + 10 cos(4.8π n ) cos(-0.4π n + 97 )

5 Final Exam Solutions ECE 223 Spring of 6 3. Sampling Applications (10 points) An intern at NOAA is given the task of designing the instrumentation for a bouy that will be located 100 miles off the coast of Cannon Beach. Among other tasks, an analog sensor on the bouy continuously measures the wave height. The wave hieght must be transmited to a base station located in Cannon Beach. The period between the waves is expected to range from 5 to 25 seconds. Express all frequencies in units of Hz. a. (1 pt) The intern decides to assume the waves are sinusoidal. What is the highest frequency component of the signal? f max = 1/5 = 0.2 Hz b. (1 pt) What is the lowest frequency component of the signal? f min = 1/25 = 0.04 Hz c. (1 pt) If the intern decides to sample this signal, what is the minimum sample rate she could use without causing aliasing? f s = 2 f max = 2/5 = 0.4 Hz Suppose a supervising engineer informs the intern that the wave height is not sinusoidal and has significant power in up to 4 harmonics. Use this model of the signal to answer the remaining questions. d. (1 pt) What is the highest frequency component of the signal given this revised model? f max = 4/5 = 0.8 Hz e. (1 pt)what is the minimum sample rate she must use? f s = 2 f max = 8/5 = 1.6 Hz f. (1 pt) Suppose the intern is concerned about high frequency noise that may be picked up by the sensor. What type of filter should she use to limit the bandwidth from 0 to f max? Lowpass Highpass Bandpass Bandstop Notch g. (1 pt) Suppose the filter must have a transition band that is at least 50% as wide as the passband. What is the passband frequency range of this filter? Passband range: 0 < f pass < f max 0 < f pass < 0.8 Hz h. (1 pt) What is the stopband frequency range of this filter? Stopband range: 1.5f max < f stop 1.2 Hz < f stop i. (1 pt) If this filter is also to be used as an anti-aliasing filter, what is the minimum sample rate that could be used with this filter? f s = 2 min(f stop ) = 2.4 Hz j. (1 pt) What is the highest frequency component in the passband of the resulting discrete-time signal. Express your answer in units of radians per sample Ω max = max(f pass ) 2π/f s = 0.8 (2π)/2.4 = π

6 Final Exam Solutions ECE 223 Spring of 6 4. Communications Concepts (8 pts) Suppose a speech signal is to be transmitted across campus using amplitude modulation techniques. You are allocated a bandwidth from 100 khz to 110 khz and limited to a transmission power of 100 watts. a. (1 pt) If you use single sideband modulation, what is the maximum bandwidth of your baseband signal? f max = 10 khz b. (1 pt) If you use double sideband modulation, what is the maximum bandwidth of your baseband signal? f max = 5 khz c. (1 pt) What type of filter must you apply to the speech signal to ensure the bandwidth of the baseband signal is within this range? Lowpass Highpass Bandpass Bandstop Notch d. (2 pts) What specifications would you use for this filter, assuming that you use double sideband modulation? Specify some reasonable passband and stopband frequencies and attenuation specifications? Note that there are many acceptable answers to this problem. Passband range: 0 < f pass < 4.5 khz Passband Attenuation: H(jω) > 0.95 Stopband range: 5 khz < f stop Stopband Attenuation: H(jω) < 0.01 e. (1 pt) If you wished to minimize the cost of the receiver, would you use synchronous amplitude modulation (SAM) or asychronous amplitude modulation (AAM)? Explain. SAM AAM Explanation: AAM only requires a few low-cost components to build a rectifier and lowpass filter needed for demodulation. SAM would require some means of obtaining or estimating the carrier frequency, which would add to the cost of the receiver. f. (1 pt) If you wished to maximize the quality of the received signal, would you use synchronous or asychronous amplitude modulation? Explain. SAM AAM Explanation: AAM uses a portion of the allocated power to transmit the carrier signal to make demodulation easier and cheaper. SAM uses all of the allocated power to transmit the modulated message signal and thereby has improved quality. g. (1 pt) What would the frequency of your carrier signal be? f = 105 khz