zt ( ) = Ae find f(t)=re( zt ( )), g(t)= Im( zt ( )), and r(t), and θ ( t) if z(t)=r(t) e

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1 Homework # Fundamentals Review Homework or EECS 562 (As needed or plotting you can use Matlab or another sotware tool or your choice) π. Plot x ( t) = 2cos(2π5 t), x ( t) = 2cos(2π5( t.25)), and x ( t) = 2cos(2π5 t ) 2 Compare these three signals and explain their similarities and dierences. 2. Let j z = + j ind: Re( z ), Im( z ), z and z and α and β i z = αe β 2. Let z(t) be a complex signal c + zt ( ) = Ae ind (t)=re( zt ( )), g(t)= Im( zt ( )), and r(t), and θ ( t) i z(t)=r(t) e j(2 π t φ) jθ () t. Let z = + j, z = j, z = + j, z = j 2 j2π t c Re( ze i ) or i =... Find 5. Find cos(2 π t)sin(2 π t) dt and what property describes the relationship between cos(2 πt) and sin(2 πt )? 6. For x( t) = 2cos( πt) sin(2 πt ) a) What is the undamental requency? b) Find the complex Fourier series or x(t). [Hint: no integration is required or this problem, convert cos( πt) and sin(2 πt ) into their complex exponential orms and note sin(α)=cos(α-π/2) and cos(α)=cos(α π)] c) Plot the double sided phase and amplitude spectrum or x(t). d) What is the power in x(t)? e) What is the bandwidth o x(t)? 7. A bit is transmitted as x(t) = cos(2πt) or a or cos(2πt) or a or ms. Find the energy and power in x(t). Ex= Px=

2 8. An input signal x(t) is processed by a ilter with an amplitude H() and phase θ() response given below. a) For x ( t) = 2cos(2π 5 t) ind output signal ya(t). a b) For x ( t) = cos(2π 75 t) ind output signal yb(t). b c) For x ( t) = 2cos(2π5 t) + cos(2π75 t) ind output signal yc(t). c d) For x ( t) = cos(2π 5 t) ind output signal yd(t). d e) For x ( t) = 2cos(2π5 t) + cos(2π5 t) ind output signal ye(t). e ) For x ( t) = 2cos(2π5 t) + cos(2π5 t) ind output signal y(t). g) Which input signal above, xa(t) x(t) has the largest bandwidth and what is that bandwidth? h) An input signal x(t) with a bandwidth B is processed by a ilter with an amplitude H() and phase θ() response given above. What is the maximum value o B that will result in distortion-less transmission o an input signal x(t) through the ilter, H()? 9. A linear time-invariant system with input signal x(t) produces an output signal y(t) =α x(t-τ), ind the system transer unction and impulse response.. The spectrum o x(t) is given below: a) The signal x(t) is sampled at 7 samples/sec to orm xs(t). Plot the spectrum o xs(t). b) For x(t) given above, what is the minimum sample rate required to prevent aliasing? c) I no aliasing is present, describe how x(t) is recovered rom xs(t). 2

3 . Two linear time invariant systems have transer unctions o H and H2 are conigured as: z(t) x(t) H() H2() y(t) x(t) H() y(t) System System 2 H and H2 have the ollowing transer unctions H ( ) = e H ( ) = + j2π j2 π (.) 2 a) Find H() such that the two systems above (System and System 2) are the same, i.e., or the same input x(t) ind H() such that System and System 2 produce the same output. b) Plot H2() c) Find h2(t). d) Find the output signal, y(t), when the input signal is x( t) = cos(2 πt). e) Is the system H() casual, Circle YES or NO, Justiy your answer. 2. An ideal bandpass ilter H() has center requency o 2 khz and bandwidth Bh= khz. The input to H() is x(t), where t kt x ( t ) = rect τ = µ = µ = τ where s and T 5 s k a) Plot H(), label axis. b) Plot X(), label axis. c) Find the power in x(t) at the undamental requency. d) For the x(t) and H() given above ind the system output y(t). [Hint: Examine the results o part a) and b) and note Y()=X()H()]

4 . Consider a linear time invariant system with a impulse response o h(t), and input signal x(t) given below. The input signal x(t) given below produces and output o y(t). x(t) exp(-t) - time h(t) time a) What is y(-5)? b) What is y(-.5)? c) What is y(.5)?. The signal x[n] is input to a LTI system with impulse response h[n]. x[n] 2 n h[n] 5 n Find the discrete time convolution o x[n]*h[n]=y[n].

5 5. A radar signal has a bandwidth o about 5 MHz. A DFT is use to analyze the requency content o a radar signal with a requency resolution o khz. a) To achieve this requency resolution what is the required record length in seconds? b) How many samples are in the record, state any assumptions? 6. Properties o the DFT. (For this problem use Matlab or another sotware tool or your choice) a) Let X[n] = cos(nπ/2), n=...6. Plot the magnitude o the DFT o X[n]. b) Let X2[n]=, n=.., cos(nπ/2), n=5..6. Plot the magnitude o the DFT o X2[n]. Explain the dierence between the results o part a) and part b). 7. Let s(t)= x(t)sin(2π o t) where o= MHz and X ( ) = rect ( ) 2 a) Plot the amplitude spectrum o s(t). b) Find the output y(t) o the ollowing system in terms o x(t). The bandwidth o the ILPF 2 is khz. [Hints: sin ( θ) = cos(2 θ) and then plot the spectrum o the signal at the 2 2 input to the ILPF] 5