GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

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1 GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMUTER ENGINEERING ECE 2026 Summer 208 roblem Set #3 Assigned: May 27, 208 Due: June 5, 208 Reading: Chapter 3 on Spectrum Representation, and Chapter 4 on Sampling and Aliasing. Quiz 2 will be in lecture on Wednesday June 3, 208. Bring your calculator. The quiz is closed book: No MATLAB or Internet; no smartphones/tablets; only one 8.5 handwritten 2-sided page of notes. A signal x( t ) is periodic if there exists a positive T such that x( t ) = x( t + T ) for all t; the fundamental period is the smallest such T. A periodic signal with fundamental period T 0 can be written as a sum of complex exponentials via the Fourier series (FS): x( t )= a k e j2 kt/t 0. k = Observe the harmonic relationship: The frequency of the k-th exponential is k times the fundamental frequency f 0 = /T 0 in Hz (or 0 = 2 /T 0 in rad/s). The k-th coefficient a k in the FS expansion can be found by analyzing one period of x( t ), according to: a k = T 0 T 0 x( t )e j2 kt/t 0 dt. ROBLEM 3..* Consider a signal of the form: x( t ) = cos(20 t) + cos(2 f 2 t), the sum of a 60-Hz sunusoid and a second sinusoid whose frequency f 2 is known to fall in the range 40 Hz f 2 50 Hz but is otherwise unspecified. This sum of sinusoids may or may not be periodic, it depends on the value of f 2. Furthermore, if it is periodic, its fundamental frequency will depend on how f 2 is related to 60 Hz. (e) (f) (g) (h) =20 Hz. =5 Hz. =2 Hz. =0 Hz. =6 Hz. =4 Hz. =2 Hz. Given an example of f 2 [40 Hz, 50 Hz] for which x( t ) is not periodic.

2 ROBLEM 3.2.* (Creating a signal with a desired instantaneous frequency.) Write an equation for a signal x( t ) in the form x( t ) = cos(. ) so that its instantaneous frequency f i ( t ), in Hz, is a linear downward sweep: from 2000 Hz at time 0, to 000 Hz at time 2. (You may want to verify your answers in MATLAB using the plotspec or spectrogram command.) Download the.wav file: Use the plotspec or spectrogram command to estimate its instantaneous frequency, then write an equation for a signal x( t ) = cos(. ) that best matches the downloaded sound. (Try to recreate the sound using a MATLAB command of the form x = cos( ). Use the soundsc command in MATLAB to listen to your recreation, and verify that it sounds similar to the original.) ROBLEM 3.3.* (Beat notes.) Consider a signal x( t ) whose two-sided spectrum is shown below: f Is this signal periodic? If the answer to part is YES, find the fundamental frequency f 0, in Hertz. If the answer to part is NO, explain why not. The signal can be written as the sum of sinusoids: x( t ) = A cos(2 f t) + A 2 cos(2 f 2 t). Find A, A 2, f, and f 2. The signal can also be written as the product of sinusoids: x( t ) = A 3 cos(2 f 3 t)cos(2 f 4 t). Find A 3, f 3, and f 4.

3 ROBLEM 3.4.* The periodic signal x( t ) shown below can be written as x( t )= x( t ) k = a k e jk t : t Find 0. (Also known as the fundamental frequency of the signal.) Find a 0. (Also known as the DC component of the signal.) ROBLEM 3.5.* Determine the Fourier series representation x( t )= the signals below. Give your answer by specifying: a k e jk2 f t for each of (i) the fundamental frequency f 0 (in Hertz); (ii) a table listing all of the nonzero Fourier series coefficients {a k }. (Hint: Although you could use the analysis formula, there is an easier way. No integration necessary!) x ( t ) = sin( t)cos( t). x 2 ( t ) = cos(2 t)cos(760 t) x 3 ( t ) = sin(0.32 t) + sin(0.36 t). ROBLEM 3.6.* Consider a periodic signal x( t ) whose two-sided spectrum is shown below: 3e j 3 3e j 3 2e j 2 2e j 2 e j Find the fundamental frequency f 0, in Hertz. Find the fundamental period T 0. Find the DC component of this signal. A periodic signal like this can be represented using a Fourier series of the form: x( t )= a k e jk2 t/t, where a k is the k-th Fourier series coefficient, k {0, ±, ±2, ±3,... }. Determine which of the Fourier series coefficients are nonzero. List these Fourier series coefficients and their values in a table.

4 ROBLEM 3.7. Answer True or False. Justify the answers with a brief explanation of your reasoning. ROBLEM 3.8. A real signal with a spectral line at 0 will necessarily have a spectral line at 0. The sum of two nonzero sinusoids can never be identically zero. The sum of two nonzero sinusoids having different frequencies can never be a single sinusoid. The sum of two periodic signals is necessarily periodic. (e) The square x 2 ( t ) of a periodic signal x( t ) is necessarily periodic. (f) If x( t ) and y( t ) are periodic with the same periodic frequency 0, then the fundamental frequency of the sum s( t ) = x( t ) + y( t ) might not be 0. s( t ) Consider the periodic 25%-duty-cycle square wave s( t ) sketched below: t ROBLEM 3.9. What is the fundamental frequency f 0 (in Hz)? Find the DC component of the signal. Find an equation for the k-th coefficient a k in the Fourier series representation s( t )= a k k = ejk2 f t. [Hint: The derivation is similar to that for the 50% duty cycle wave of Sect ] Define s 3 ( t ) as the signal synthesized using only the DC term plus the first three harmonics. Write an equation for s 3 ( t ) as a constant plus sinusoids. [You will probably want to check your work by generating and plotting s 3 ( t ) in MATLAB and verifying that it approximates s( t ).] Consider the signal x( t ) = 3 + 4cos(45 t ) + 5cos(75 t 0.35 ). Sketch the two-sided spectrum for x( t ). Find the fundamental period T 0 (in seconds). Find the fundamental frequency f 0 (in Hz). A periodic signal like this can be represented using a Fourier series of the form: x( t )= a ke jk2 t/t, where a k is the k-th Fourier series coefficient, k { 0, ±, ±2, ±3,... }. List all of the nonzero Fourier series coefficients and their values in a table.

5 2 ROBLEM 3.0. Suppose x( t ) = k = 2 (k2 + )e j2k t. Is this signal periodic? If yes, specify the fundamental frequency f 0 (in Hz). If not, explain why not. Sketch the two-sided spectrum of x( t ). What is the DC component of x( t )? ROBLEM 3.. (The impact of delay on the Fourier series coefficients.) If a periodic signal s( t )= a k e jk2 t/t is delayed by half of its period, yielding y( t ) = s(t T 0 /2), then the resulting signal will still be periodic with period T 0, meaning that it too will have a Fourier series representation: y( t )= k = b k e jk2 t/t The question is: How do the Fourier coefficients {b k } of the delayed signal relate to the Fourier series coefficient {a k } of the original signal? In other words, find an equation for b k expressed as a function of a k.