problem 1: Sketch the Fourier transform of a rectangular pulse of amplitude 10 V and width 0.1 second that is centered on the zero time axis. Indicate the amplitude at dc and for the first spectral peak. Also indicate the frequency of the first spectral peak and points of zero amplitude.
problem 2: The signal exp(-3t) u(t) is delayed 0.1 seconds and subtracted from itself. What is the Fourier Transform of the output signal? Draw a block diagram of the system. What is the Transfer Function of the system, H(f)?
problem 3: What is the autocorrelation function of a rectangular pulse with an amplitude of 5 volts that exists between t = 0 and t = 10 seconds? Sketch your answer (a graphical solution is OK).
problem 4: A signal, x(t) = 2 sin (10π 10^{6} t), is applied to the input of an RC lowpass filter with a cutoff frequency of 1 MHz. What is the output signal? How many dB is the output signal relative to the input signal?
problem 5: Find out the instantaneous frequency of a signal v(t) = 5 cos (10πt + cos 5πt) at t = 10 seconds.
problem 6: A 100 MHz carrier is frequency modulated by a 1 kHz sinusoidal signal and has a peak frequency deviation of 2 kHz. Draw the spectrum of this signal. Show the frequency and amplitude (relative to the unmodulated carrier level) of at least the first 2 sidebands.
problem 7: Sketch the constellation patterns (magnitude and phase) of a BPSK, a QPSK and an 8PSK digitally modulated signal. If all three types of signals occupy the same bandwidth, what is the ratio between their relative maximum information transmission rates?