145M Final Exam Solutions page 1 May 13, 2006 S. Derenzo
UNIVERSITY OF CALIFORNIA
College of Engineering
Electrical Engineering and Computer Sciences Department
145M Microcomputer Interfacing Lab
Final Exam Solutions May 13, 2006
1.1 Tri-State Buffer: Digital circuit with input and output lines, and a select line. When the
select line is in one logic state, output = input. Otherwise the output is in a high-impedance
state and neither drives nor loads any circuit connected to it.
1.2 Fourier Convolution Theorem: The Fourier transform of the convolution of two
functions is the simple product of the Fourier transforms of the two functions
[2 points off for stating the Fourier Frequency Convolution Theorem]
1.3 Transition Voltages (A/D Converter): Specific input voltages at which the output
switches between one output number and the next.
1.4 Infinite Impulse Response Digital Filter: A digital filter whose output depends on
previous outputs
Note: The above definitions were taken directly from the textbook.
2.1
r=a/b∂r/∂a=1 / b∂r/∂b=−a/b2
σ
r
2=
σ
a
2(∂f/∂a)2+
σ
b
2(∂f/∂b)2=
σ
a
2(1/ b2)+
σ
b
2(a2/b4)
σ
r
2=10−4a2(1 / b2)+10−4b2(a2/b4)=2×10−4(a2/b2)
σ
r=0.01414( a/b)=0.01414 r
3.1 A periodic waveform is the convolution of a time-limited waveform and a comb of delta
functions. The Fourier transform of a periodic waveform is the simple product of the
Fourier transform of the time-limited waveform and the Fourier transform of a comb of delta
functions. The latter is a comb of delta functions in the frequency domain so the simple
product is nonzero only at discrete frequencies. See Figure 5.26 of the textbook.
[7 points off if the periodic function is not described as the convolution of a time-limited
function and a train of delta functions]
3.2 Sampling in the time domain is equivalent to a simple product of the waveform and a comb
of delta functions in the time domain. The Fourier transform of the sampled data is the
convolution of the Fourier transform of the wavefunction and the Fourier transform of the
comb of delta functions. Since the latter is a comb of delta functions in the frequency
domain, overlap will occur unless the original waveform is not frequency limited. This
overlap causes aliasing and can be eliminated by increasing the sampling frequency or by
bandwidth limiting the waveform so that the highest frequencies persent are sampled at least
twice per cycle. See Figure 5.29 of the textbook.
[10 points off for only stating the Nyquist Theorem]
[5 points off for stating that aliasing is caused by frequency overlap but not stating that
sampling is equivalent to multiplying by a train of delta function in the time domain]
