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Spring 1994, EECS 121 -- Midterm 1 University of California College of Engineering Department of Electrical Engineering and Computer Science Professor Wong Spring 1994
EECS 121 -- MIDTERM 1
(Closed book and notes)
1. Consider a linear and time-invariant system (denoted by A ) for which an input x ( t ) = cos2(pi) ft produces an output y ( t ) = 1 / (1 + (2(pi) f )^2) * (cos2(pi) ft + (2(pi) f )sin2(pi) ft ) Find the output y ( t ) in each of the following situations. 2. Consider a narrowband signal x ( t ) = A ( t ) cos (200(pi) t + (theta)( t )) where A ( t ) = ( (sin(pi) t / ((pi) t ) )^2 and (theta)( t ) = 2(pi)sin2(pi) t. Find the output y ( t ) of each of the following systems where x ( t ) is the input:
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Spring 1994, EECS 121 -- Midterm 1 The filter transfer functions are given by H 1 ( f ) = 1 , | f | <= 1 ; = 0 otherwise H 2 ( f ) = 1 , if | f - 100| <= 1 or | f + 100| <= 1 ; = 0 otherwise
3. Let X be a random variable with a probabilty density function given by P (^) X = 1/2 | x | <= 1 ; = 0 | x | > 1
a. Find E | X | b. Let Y be another random variable whose conditional density given X is p ( y | x ) = 1 / root(2(pi)) * e ^(-1/2 * ( y - x )^2) Find EXY.
4. For each of the following functions R ( t , s ), determine whether it can be an autocorrelation function. Explain. a. R ( t , s ) = e ^( -| t - s | ) * cos20(pi)( t - s ) b. R ( t , s ) = cos^2( t + s ) c. R ( t , s ) = 1 - t - s + ts , 0 <= s , t <= 1 5. Let X (^) t have a power spectral density function
P x ( f ) = 1 , | f | <= 1 ; = 0 otherwise
a. Find the autocorrelation function Rx (tau). b. Let Xt be the input and Y (^) t the output to a linear and time invariant system with transfer function
H ( f ) = e ^(-| f |) , -infinity < f < infinity Find the average power of the output E | Y (^) t |^.
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