MECH 322 Instrumentation

Performed: 03/15/06

Differentiation and Spectral Analysis of Discretely Sampled Signals

Group 0

Pablo Araya

Lab Instructors:Mithun Gudipati, Venkata Venigalla

ABSTRACT

• The objective of this lab is to measure sine and sawtooth wave voltage signals using a data acquisition system and demonstrate errors that can occur when determining their time derivative and spectral content.

• Time derivatives were determined using first order finite differencing with different time steps. The random noise decreased as the time step increased but the accuracy of following step changes in slope decreased.

• The peak in the measured spectral content was in agreement with signal frequency when the sampling frequency was at least twice the signal frequency.

Table. 1 Sine and Sawtooth Wave Parameters

• Peak to peak voltage and maximum frequency are determined from scope and frequency counter.

• The maximum slopes of the ideal wave forms are (dV/dt)max,sine=VPPfM,and (dV/dt)max,sawtooth = 2VPPfM

• Input resolution error Wv depends on A/D converter voltage range (±5V) and number of bits (N = 14) and Wv = 0.000305

Waveform VPP [volts] fM [Hz] dV/dtIdeal,Max [V/s]Sine 1.6 100 503

Sawtooth 1.6 100 320

Fig.1 Sine Wave and Derivative Based on Different Time Steps

• dV/dt1 (t=0.0000208 sec) is nosier than dV/dt10 (t=0.000208 sec)• The maximum slope from the finite difference method is larger than

the ideal value. This may be because the actual wave was not a pure sinusoidal.

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dV

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lts/

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V [

Vo

lts]

t [sec]

dV/dtIdeal,Min

dV/dtIdeal,Max

V(t)

dV/dtm=1

dV/dtm=10

Fig. 2 Sawtooth Wave and Derivative Based on Different Time Steps

• dV/dt1 is again nosier than dV/dt10

• dV/dt1 responds to the step change in slope more accurately than dV/dt10

• The maximum slope from the finite difference method is larger than the ideal value.

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dV

/dt [

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lts/

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V [

Vo

lts]

t [sec]

dV/dtIdeal,Min

dV/dtIdeal,MaxV(t)

dV/dtm=1 dV/dtm=10

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frecuency f [Hz]

VR

MS [

Vo

lts]

fs = 300 Hz fs = 5000 Hz

fs = 150 Hzfs = 70 Hz

Fig. 3 Measured Spectral Content of 100 Hz Sine Wave for Different Sampling Frequencies

• The measured peak frequency fP equals the maximum signal frequency fM = 100 Hz when the sampling frequency fS is greater than 2fM

• fs = 70 and 150 Hz do not give accurate indications of the peak frequency.

Table 2 Peak Frequency versus Sampling Frequency

• For fS > 2fM = 200 Hz the measured peak is close to fM.

• For fS < 2fM the measured peak is close to the magnitude of fM–fS.

• The results are in agreement with sampling theory.

Sampling Frequency, fs [Hz] 5000 300 150 70

Peak Spectral Frequency, fp [Hz] 100 100 50 30

Fig. 4 VI Front Panel

Figure 5 VI Block Diagram

Statistics This Express VI produces the following measurements: Time of Maximum

Convert to Dynamic Data Double Click the Icon and select “Single Waveform” (it is located at the bottom of the list)

Convert from Dynamic Data Double Click the Icon and select “1D array of scalars – single waveform”

Table 3 Signal and Indicated frequency data (extra credit)

• This table shows the dimensional and dimensionless signal frequency fm (measured by scope) and frequency indicated by spectral analysis, fa.

• For a sampling frequency of fS = 48,000 Hz, the folding frequency is fN = 24,000 Hz.

fm [Hz] fa [Hz] fm/fN fa/fN0 0 0.00 0.00

9910 9925 0.41 0.4119540 19575 0.81 0.8223120 23125 0.96 0.9630190 17800 1.26 0.7440510 7475 1.69 0.3147320 675 1.97 0.0350180 2175 2.09 0.0961200 13275 2.55 0.5571800 23850 2.99 0.9972400 23575 3.02 0.9879800 16125 3.33 0.6789500 6475 3.73 0.2795400 475 3.98 0.0299700 3725 4.15 0.16

Figure 6 Dimensionless indicated frequency versus signal frequency (extra credit)

• The characteristics of this plot are similar to those of the textbook folding plot

• For each indicated frequency fa, there are many possible signal frequencies, fm.

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f a/f

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fm/fN