Chapter 13
Fourier Analysis Techniques

m13.3 Fourier Transform

For the waveform shown in Figure m13.3:

1. Determine its Fourier transform.
2. Plot the amplitude spectrum |F(ω)| with MathScript or MATLAB as follows:
   1. Frequency 0 ≤ f ≤ 4000 Hz (remember to convert angular frequency ω to oscillation frequency f),
   2. A = 10, and
   3. τ = 1, 2, and 4 ms (create three distinct plots).
3. Determine the frequency at which the first null occurs in each of the three plots.
4. Discuss the relationship between the rectangular pulse width and the width of the main lobe of the amplitude spectrum.

NI Multisim Measurements

1. Create a circuit with a PULSE_VOLTAGE source. Set the “Period” to 100 ms; set the remaining parameters as needed to create the pulse shown in Figure m13.3 with A = 10. Note that the pulse must shift right to begin at t = 0; this shift does not affect the amplitude spectrum.
2. Plot the amplitude line spectrum of f(t) with a Simulate → Analyses → Fourier Analysis for τ = 1, 2, and 4 ms (create three plots). Set the following parameter values:
   • “Frequency Resolution (fundamental frequency)” = 10 Hz,
   • “Number of harmonics” = 400,
   • “Display” = “Graph,” and
   • “Vertical scale” = “Linear.”
3. Determine the frequency at which the first null occurs in each of the three plots.

NI myDAQ Measurements

1. Set up your myDAQ and ELVISmx instruments as follows:
   1. Connect myDAQ Analog Output 0 to Analog Input 0, i.e., AO0 to AI0+ and AGND to AI0-.
   2. Create the rectangular pulse f(t) with the ELVISmx Function Generator in squarewave mode. Set the frequency to 10 Hz. Adjust the amplitude and DC offset controls to match the pulse waveform shown in Figure m13.3 with A = 10. Control the pulse width with the “Duty Cycle” control.
2. Plot the power spectrum of f(t) on the ELVISmx Dynamic Signal Analyzer (DSA) for τ = 1, 2, and 4 ms (create three plots). Adjust the panel controls to match the following settings:
   • FFT Settings:
     1. Frequency Span = 4000
     2. Resolution (lines) = 400
     3. Window = None
   • Averaging:
     1. Mode = None
   • Frequency Display:
     1. Units = Linear
     2. Mode = Peak
   • Scale Settings:
     1. Scale = Auto
3. Determine the frequency at which the first null occurs in each of the three plots.

Further Exploration with NI myDAQ

1. Frequency spectrum plots normally possess much higher dynamic range than their corresponding time-domain plots. Review the DSA plots you created for the widest rectangular plot (τ = 4 ms), especially the side lobe amplitudes beyond the first null frequency. Note how their values appear quite small compared to the amplitude of the main lobe. Now set the “Frequency Display” units to “dB” (decibels); you can stabilize the display by setting the “Scale Settings” from “Auto” to “Manual.” Discuss the merits of a logarithmic display scale compared to a linear display scale.
2. To further experience the advantages of a logarithmic display, repeat the experiment with a single sinusoidal component. Set the function generator to sinusoidal mode at 500 Hz and remove the DC offset. Display the spectrum with “Linear” units and then with “dB” units. Stabilize and improve the measurement by setting the “Averaging” mode to “RMS.” Discuss your observations.