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ENEE 420 Due 10/08/2002 Homework 05 1. A zero-mean white Gaussian noise process with power spectral density N0 is passed through 2 an ideal lter...

2. Consider the coherent demodulator for DSB-SC AM when the receiver has a phase mismatch

where Nz(t) is our complex Gaussian noise process, the phase error is e =  ô€€€ ^, and
Yz(t) = Acm(t)ej + Nz(t)
(a) Assume e is constant and derive an expression for the output ^m(t).
(b) Determine the SNR as a function of e.
(c) Determine the average SNR if e is uniformly distributed
ENEE 420 Due 10/08/2002 Homework 05 1. A zero-mean white Gaussian noise process with power spectral density N 0 2 is passed through an ideal ﬁlter whose passband is from 10 kHz 18 kHz. The output process is denoted N c ( t ). S (f) N c 10 18 -10 -18 N /2 0 f (kHz) (a) If f c = 14 kHz, as it might be for DSB, ﬁnd and sketch S N I ( f ) and S N I N Q ( f ). (b) If f c = 12 kHz, as it might be in VSB, ﬁnd and sketch S N I ( f ) and S N I N Q ( f ). 2. Consider the coherent demodulator for DSB-SC AM when the receiver has a phase mismatch Re( ) V (t) z Y (t) z H (f) L V (t) I m(t) exp(-j ϕ ) where N z ( t ) is our complex Gaussian noise process, the phase error is φ e = φ - ˆ φ , and Y z ( t ) = A c m ( t ) e + N z ( t ) (a) Assume φ e is constant and derive an expression for the output ˆ m ( t ). (b) Determine the SNR as a function of φ e . (c) Determine the average SNR if φ e is uniformly distributed on h - π 2 , π 2 i . 3. A white noise process, W ( t ) is added to a signal, x ( t ), with power spectral density, S x ( f ). The sum is passed through an ideal lowpass ﬁlter with unity passband gain and bandwidth B where B > W . Determine the SNR at the ﬁlter output. By what factor will the SNR increase if B is reduced to W . S W ( f ) = N 0 2 S x ( f ) = ( Af 2 | f | < W 0 elsewhere 1
4. This problem explores the performance of VSB AM modulation. The transmitter is X (t) z H (f) z m(t) A c where H z ( f ) = 2 W 2 < f < W f + W/ 2 W/ 2 - W 2 < f < W 2 0 elsewhere W W/2 0 -W/2 -W H (f) z 2 1 f (a) Determine and sketch H I ( f ) and H Q ( f ). Using these, explain why this model is equiva- lent to Fig. 5.17 in Fitz. (b) Show that the standard coherent demodulator will still work. Re( ) V (t) z Y (t) z H (f) L V (t) I m(t) exp(-j ϕ ) (c) Characterize the output noise N L ( t ) and determine the noise power if the receive band- pass ﬁlter, H R ( f ), is just wide enough to pass the transmitted signal. (d) Determine the output SNR. If necessary assume the message signal is a random process with power spectral density S M ( f ) = ( 1 | f | < W 0 | f | > W (e) To verify that your answer makes sense, use the following H z ( f ) to obtain an expression for SNR as a function iof f v , then consider what happens when f v = 0. H z ( f ) = 2 f v < f < W f + f v f v - f v < f < f v 0 elsewhere W fv 0 -fv -W H (f) z 2 1 f and consider what happens when f v = 0 and when f v gets very large. (f) Now use the following H z ( f ) and consider what happens as f v → ∞ . H z ( f ) = f + f v f v | f | < f v 0 elsewhere W fv 0 -fv -W H (f) z 2 1 2
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