#### 4. (20 points) Consider the periodic signal x(n) = {1″, 0, 0, 1}, where x(n) is periodic in n with period N = 4. (a) Find the Fourier series representation of x(n) (b) Let Px denote the average power of x(n). Find Px. (c) plot the magnitude |ck| and the phase ✓k of the Fourier coefficient versus k. (d) Is the sequence of the Fourier coefficients {ck} periodic in k? if yes, what is the period of this sequence?

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#### 3. (20 points) Consider the causal and BIBO-stable LTI system described by the following first order constant coincide deference equation: (a) Determine the system function H(z) and the impulse response h(n), by taking the inverse Z-transform of H(z). (b) Suppose the system is initially relaxed, is applied to the relaxed system. Determine the output y(n) (note that y(n) is the zero-state response of the system). Determine the natural response of the system ynr(n), and the forced response of the system yfr(n). (c) Suppose the system is non-relaxed is applied to the non-relaxed system. Determine the zero-input response of the system yzi(n), the zero-state response of the system yzs(n), and the total response of the system y(n). • (Bonus Problem, 5 points) Based on the pole-zero plot and the ROC of H(z), what type of

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#### Question 2 The following BJT amplifier circuit will amplify a 12mV sinusoidal signal from a microphone to a 0.6V sinusoidal output signal. Operating Q-point is Ic = 0.2 mA, and V CEQ = 4V . Assume the output resistance of the microphone is 0.5 KW (as shown) . Consider a transistor used has b = 100 and V BE = 0.7V . Determine Rc

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#### 6. [Extra Credit] Let x(t)be a real-valued signal carrying the message that we want to deliver, for which . A communication transmitter performs the amplitude modulation to produce the signal The signal g (t)travels from the transmitter to a receiver. Here, we assume there is no path loss or phase distortion, i.e., the signal g(t) is received by the receiver perfectly. At the receiver, the received signal g(t)is demodulated, shown in the following figure. Here, symbol denotes the multiplication operation. The ideal lowpass filter has cutoff frequency of 2,000π and passband gain of 2. Determine the demodulator output y(t). Please list the detail steps to receive full credit [Extra Credit 10 points].

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#### 5. The signal y(t)is generated by convolving a band-limited signal(t) x1(t) with another band-limited signal x2(t) , that is where * stands for the convolution, and Impulse-train sampling is performed on y t( )to obtain Specify the range of values for the sampling period T which ensures that y(t) is recoverable from yp(t) . [15 points]

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