145M April 16, 2008 page 1 S. Derenzo
`Solutions for Midterm #2 - EECS 145M Spring 2008
1.1 Frequency aliasing is caused by sampling at a frequency fs that is less than twice the highest
frequency fmax in the waveform.
It may be avoided by using an analog low-pass filter to eliminate frequencies above fs/2.
[4 points off for increasing the sampling frequency, since fmax is not known]
1.2 Spectral leakage is caused when frequency components are not sampled for a whole number
of cycles, which results in a discontinuity between the last sample and the “next sample”,
which is also the first sample.
It is avoided by multiplying the sampled values by a windowing function that has zero value
and zero slope at the ends of the sampling window.
[3 points off for a solution that samples for a longer time- this does not eliminate the
discontinuity and long-range spectral leakage will still occur]
[3 points off if the only answer is to sample for an integer number of cycles- nothing is
known about the frequency components present]
2.1 H(f) can only be non-zero at multiples of the repeat frequency = n/P
[4 points off for describing or sketching only the first harmonic]
2.2 All the components in the Fourier integral have periods that are integer multiples of P, so
integrating from 0 to P is all that is necessary.
H(f)=h(t)g(t)dt g(t)=exp(−j2
π
nt /P)
∫
Multiply and integrate
h(t)
g(t) g(t+P/2) = g(t)
h(t+P/2) = -h(t)
0P/2 P
g(t) = exp(-j2!nt/P)
For n even, g(t+P/2) = exp(-j2!n[t+P/2]/P) = exp(-j2!nt/P-j2!n/2) = exp(-j2!nt/P) = g(t)
For even n, the harmonic factor exp(-j2!nt/P) is the same in the interval from t = 0 to P/2 as it is
in the interval from t = P/2 to P. However, h(t) has the opposite sign from t =0 to P/2 as it does
from t = P/2 to P and the two inner products cancel.
[Full credit for showing that the integral from 0 to P could be split into two integrals that
cancel][4 exams out of 17 had part 2.2 correct]
[5 points off for a valiant but incomplete attempt]
[the most common approach was to integrate both h(t)g(t) and h(t+P/2)g(t+P/2) from 0 to P,
without following up with a convincing argument that each was zero for even n.]
