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c. Based on (a) and (b), sketch a plot of the expected amplitude spectrum. What is the frequency
spacing on the abscissa? What is its Nyquist frequency?
7.30 An electrical signal from a transducer is sampled at 20,000 Hz using a 12-bit successive
approximation A/D converter. A total of 5 seconds of data are acquired and passed to computer
memory. How much 16-bit computer memory (in kbytes) is required?
7.31 How many data points can be sampled by passing a signal through a 12-bit parallel A/D converter and
stored in computer memory, if 8 MB of 32-bit computer memory is available? If the A/D converter uses
a 100-MHz clock and acquisition is by DMA, estimate the duration of signal that can be measured.
7.32 Select an appropriate sample rate and data number to acquire with minimal leakage the first five
terms of a square wave signal having a fundamental period of 1 second. Select an appropriate cutoff
frequency for an anti-alias filter. Hint: Approximate the square wave as a Fourier series.
7.33 A triangle wave with a period of 2 seconds can be expressed by the Fourier series
y t
ð Þ ¼
X
2D 1 ð1 À cos npÞ=np
½
cos pnt n ¼ 1; 2; . . .
Specify an appropriate sample rate and data number to acquire the first seven terms with minimal
leakage. Select an appropriate cutoff frequency for an anti-alias filter.
7.34 Using Fourier transform software (or equivalent software), generate the amplitude spectrum for the
square wave of Problem 7.32.
7.35 Using Fourier transform software (or equivalent software), generate the amplitude spectrum for the
triangle wave of Problem 7.33. Use D 1 ¼ 1 V.
7.36 A single-stage low-pass RC Butterworth filter with f c ¼ 100 Hz is used to filter an analog signal.
Determine the attenuation of the filtered analog signal at 10, 50, 75, and 200 Hz.
7.37 A three-stage LC Bessel filter with f c ¼ 100 Hz is used to filter an analog signal. Determine the
attenuation of the filtered analog signal at 10, 50, 75, and 200 Hz.
7.38 Design a cascading LC Butterworth low-pass filter that has a magnitude ratio flat to within 3 dB from
0 to 5 kHz but with an attenuation of at least 30 dB for all frequencies at and above 10 kHz.
7.39 Choose an appropriate cascading low-pass filter to remove a 500-Hz component contained within an
analog signal that is to be passed through an 8-bit A/D converter having 10-V range and 200-Hz
sample rate. Attenuate the component to within the A/D converter quantization error.
7.40 The voltage output from a J-type thermocouple referenced to 0
C is to be used to measure temperatures
from 50 to 70
C. The output voltages vary linearly over this range from 2.585 to 3.649 mV.
a. If the thermocouple voltage is input to a 12-bit A/D converter having a Æ5-V range, estimate the
percent quantization error in the digital value.
b. If the analog signal can be first passed through an amplifier circuit, compute the amplifier gain
required to reduce the quantization error to 5% or less.
c. If the ratio of signal-to-noise level (SNR) in the analog signal is 40 dB, compute the magnitude of
the noise after amplification. Discuss the results of 7.40b in light of this.
7.41 Specify an appropriate Æ5-V M-bit A/D converter (8- or 12-bit), sample rate (up to 100 Hz) and
signal conditioning to convert these analog signals into digital series. Estimate the quantization error
and dynamic error resulting from the system specified:
a. E(t) ¼ 2 sin 20pt V
b. E(t) ¼ 1.5 sin pt þ 20 sin 32pt À 3 sin (60pt þ p/4) V
c. P(t) ¼ À 10 sin 4pt þ 5 sin 8pt psi; K ¼ 0.4 V/psi
Problems 307
14:43:59 Page 307
c. Based on (a) and (b), sketch a plot of the expected amplitude spectrum. What is the frequency
spacing on the abscissa? What is its Nyquist frequency?
7.30 An electrical signal from a transducer is sampled at 20,000 Hz using a 12-bit successive
approximation A/D converter. A total of 5 seconds of data are acquired and passed to computer
memory. How much 16-bit computer memory (in kbytes) is required?
7.31 How many data points can be sampled by passing a signal through a 12-bit parallel A/D converter and
stored in computer memory, if 8 MB of 32-bit computer memory is available? If the A/D converter uses
a 100-MHz clock and acquisition is by DMA, estimate the duration of signal that can be measured.
7.32 Select an appropriate sample rate and data number to acquire with minimal leakage the first five
terms of a square wave signal having a fundamental period of 1 second. Select an appropriate cutoff
frequency for an anti-alias filter. Hint: Approximate the square wave as a Fourier series.
7.33 A triangle wave with a period of 2 seconds can be expressed by the Fourier series
y t
ð Þ ¼
X
2D 1 ð1 À cos npÞ=np
½
cos pnt n ¼ 1; 2; . . .
Specify an appropriate sample rate and data number to acquire the first seven terms with minimal
leakage. Select an appropriate cutoff frequency for an anti-alias filter.
7.34 Using Fourier transform software (or equivalent software), generate the amplitude spectrum for the
square wave of Problem 7.32.
7.35 Using Fourier transform software (or equivalent software), generate the amplitude spectrum for the
triangle wave of Problem 7.33. Use D 1 ¼ 1 V.
7.36 A single-stage low-pass RC Butterworth filter with f c ¼ 100 Hz is used to filter an analog signal.
Determine the attenuation of the filtered analog signal at 10, 50, 75, and 200 Hz.
7.37 A three-stage LC Bessel filter with f c ¼ 100 Hz is used to filter an analog signal. Determine the
attenuation of the filtered analog signal at 10, 50, 75, and 200 Hz.
7.38 Design a cascading LC Butterworth low-pass filter that has a magnitude ratio flat to within 3 dB from
0 to 5 kHz but with an attenuation of at least 30 dB for all frequencies at and above 10 kHz.
7.39 Choose an appropriate cascading low-pass filter to remove a 500-Hz component contained within an
analog signal that is to be passed through an 8-bit A/D converter having 10-V range and 200-Hz
sample rate. Attenuate the component to within the A/D converter quantization error.
7.40 The voltage output from a J-type thermocouple referenced to 0
C is to be used to measure temperatures
from 50 to 70
C. The output voltages vary linearly over this range from 2.585 to 3.649 mV.
a. If the thermocouple voltage is input to a 12-bit A/D converter having a Æ5-V range, estimate the
percent quantization error in the digital value.
b. If the analog signal can be first passed through an amplifier circuit, compute the amplifier gain
required to reduce the quantization error to 5% or less.
c. If the ratio of signal-to-noise level (SNR) in the analog signal is 40 dB, compute the magnitude of
the noise after amplification. Discuss the results of 7.40b in light of this.
7.41 Specify an appropriate Æ5-V M-bit A/D converter (8- or 12-bit), sample rate (up to 100 Hz) and
signal conditioning to convert these analog signals into digital series. Estimate the quantization error
and dynamic error resulting from the system specified:
a. E(t) ¼ 2 sin 20pt V
b. E(t) ¼ 1.5 sin pt þ 20 sin 32pt À 3 sin (60pt þ p/4) V
c. P(t) ¼ À 10 sin 4pt þ 5 sin 8pt psi; K ¼ 0.4 V/psi
Problems 307