E1C07 09/14/2010
14:43:59 Page 305
PROBLEMS
For many of these problems, using spreadsheet software or the accompanying software will facilitate
solution.
7.1 Convert the analog voltage E(t) ¼ 2 sin 4pt mV into a discrete time signal. Specifically, use sample
time increments of (a) 1/8 second, (b) 1/5 second, (c) 1/3 second, and (d) 1/21 second, and use a data
set of 128 points. Plot 4 seconds of each series as a function of time. Discuss apparent differences
between the discrete representations of the analog signal.
7.2 For the analog voltage E(t) ¼ 2 sin 4pt mV sampled at time increments of (a) 1/8 second,
(b) 1/5 second, (c) 1/3 second, and (d) 1/21 second, you are to compute the DFT for each. Use a data
set of 128 points. Discuss apparent differences.
7.3 An analog signal has the form E(t) ¼ 4 sin 8pt (V). Compute the DFT at sample rates of 4 Hz and
16 Hz. Use a data set of 256 points. Discuss and compare your results.
7.4 Determine the alias frequency that results from sampling f 1 at sample rate f s :
a.
f 1 ¼ 60 Hz; f s ¼ 90 Hz
c.
f 1 ¼ 10 Hz; f s ¼ 6 Hz
b.
f 1 ¼ 1.2 kHz; f s ¼ 2 kHz
d.
f 1 ¼ 16 Hz; f s ¼ 8 Hz
7.5 A particular data-acquisition system is used to convert the analog signal E(t) ¼ (sin 2pt þ
2 sin 8pt) V into a discrete time signal using a sample rate of 16 Hz. Build the discrete time
signal and from that use the Fourier transform to reconstruct the Fourier series.
7.6 Consider the continuous signal found in Example 2.3. What would be an appropriate sample rate and
sample period to use in sampling this signal if the resulting discrete series must have a size of 2
M ,
where M is an integer and the signal is to be filtered at and above 2 Hz?
7.7 Convert the analog signal E(t) ¼ (4 þ 2 sin 4pt þ 3 sin 16pt) V into a discrete time signal using
a sample rate of 32 Hz. Build the discrete time signal and its amplitude and phase spectra. Then try at
f s ¼ 16 Hz and at f s ¼ 40 Hz. Discuss results.
7.8 Convert the following straight binary numbers to positive integer base 10 numbers:
a.
1010
c.
10111011
b.
11111
d.
1100001
7.9 Convert: (a) 1100111.1101 (binary) into a base 10 number; (b) 4B2F into straight binary; (c) 278.632
(base 10) into straight binary.
7.10 Convert the following decimal (base 10) numbers into bipolar binary numbers using a twos
complement code:
a.
10
c.
À247
b.
À10
d.
1013
7.11 A computer does integer arithmetic in twos complement binary code. How is the largest positive
binary number represented in this code for an 8-bit byte? Add one to this number. What base 10
decimal numbers do these represent?
7.12 How is the largest negative binary number represented in twos complement code for an 8-bit byte.
Subtract one from this number. What base 10 decimal numbers do these represent?
7.13 List some possible sources of uncertainty in the dual-slope procedure for A/D conversion. Derive a
relationship between the uncertainty in the digital result and the slope of the integration process.
7.14 Compute the resolution and SNR for an M-bit A/D converter having a full-scale range of Æ5 V. Let M
be 4, 8, 12, and 16.
Problems 305
14:43:59 Page 305
PROBLEMS
For many of these problems, using spreadsheet software or the accompanying software will facilitate
solution.
7.1 Convert the analog voltage E(t) ¼ 2 sin 4pt mV into a discrete time signal. Specifically, use sample
time increments of (a) 1/8 second, (b) 1/5 second, (c) 1/3 second, and (d) 1/21 second, and use a data
set of 128 points. Plot 4 seconds of each series as a function of time. Discuss apparent differences
between the discrete representations of the analog signal.
7.2 For the analog voltage E(t) ¼ 2 sin 4pt mV sampled at time increments of (a) 1/8 second,
(b) 1/5 second, (c) 1/3 second, and (d) 1/21 second, you are to compute the DFT for each. Use a data
set of 128 points. Discuss apparent differences.
7.3 An analog signal has the form E(t) ¼ 4 sin 8pt (V). Compute the DFT at sample rates of 4 Hz and
16 Hz. Use a data set of 256 points. Discuss and compare your results.
7.4 Determine the alias frequency that results from sampling f 1 at sample rate f s :
a.
f 1 ¼ 60 Hz; f s ¼ 90 Hz
c.
f 1 ¼ 10 Hz; f s ¼ 6 Hz
b.
f 1 ¼ 1.2 kHz; f s ¼ 2 kHz
d.
f 1 ¼ 16 Hz; f s ¼ 8 Hz
7.5 A particular data-acquisition system is used to convert the analog signal E(t) ¼ (sin 2pt þ
2 sin 8pt) V into a discrete time signal using a sample rate of 16 Hz. Build the discrete time
signal and from that use the Fourier transform to reconstruct the Fourier series.
7.6 Consider the continuous signal found in Example 2.3. What would be an appropriate sample rate and
sample period to use in sampling this signal if the resulting discrete series must have a size of 2
M ,
where M is an integer and the signal is to be filtered at and above 2 Hz?
7.7 Convert the analog signal E(t) ¼ (4 þ 2 sin 4pt þ 3 sin 16pt) V into a discrete time signal using
a sample rate of 32 Hz. Build the discrete time signal and its amplitude and phase spectra. Then try at
f s ¼ 16 Hz and at f s ¼ 40 Hz. Discuss results.
7.8 Convert the following straight binary numbers to positive integer base 10 numbers:
a.
1010
c.
10111011
b.
11111
d.
1100001
7.9 Convert: (a) 1100111.1101 (binary) into a base 10 number; (b) 4B2F into straight binary; (c) 278.632
(base 10) into straight binary.
7.10 Convert the following decimal (base 10) numbers into bipolar binary numbers using a twos
complement code:
a.
10
c.
À247
b.
À10
d.
1013
7.11 A computer does integer arithmetic in twos complement binary code. How is the largest positive
binary number represented in this code for an 8-bit byte? Add one to this number. What base 10
decimal numbers do these represent?
7.12 How is the largest negative binary number represented in twos complement code for an 8-bit byte.
Subtract one from this number. What base 10 decimal numbers do these represent?
7.13 List some possible sources of uncertainty in the dual-slope procedure for A/D conversion. Derive a
relationship between the uncertainty in the digital result and the slope of the integration process.
7.14 Compute the resolution and SNR for an M-bit A/D converter having a full-scale range of Æ5 V. Let M
be 4, 8, 12, and 16.
Problems 305