Chapter 13
Fourier Analysis Techniques

m13.1 Fourier Series Representation

Consider the voltage waveform v(t) shown in Figure m13.1.

1. Determine if the waveform has dc, even, or odd symmetry.
2. Obtain its cosine/sine Fourier series representation.
3. Convert the representation to amplitude/phase format and plot the amplitude line spectrum for n = 0 to 5 using A = 10 volts and T = 4 ms.

NI Multisim Measurements

1. Create the voltage waveform v(t) of Figure m13.1 with a PIECEWISE_LINEAR_VOLTAGE source. Use the same amplitude and period as in the problem statement.
2. Plot and tabulate the amplitude line spectrum of v(t) with a Simulate → Analyses → Fourier Analysis :
   1. Set the “Frequency Resolution (fundamental frequency)” parameter to match the fundamental frequency f₀ of the voltage waveform v(t).
   2. Leave the remaining parameters at their default settings.

NI myDAQ Measurements

1. Connect myDAQ Analog Output 0 to Analog Input 0, i.e., AO0 to AI0+ and AGND to AI0-.
2. Create the voltage waveform v(t) with the ELVISmx Arbitrary Waveform Generator using the same amplitude and period as in the problem statement. Set the sampling frequency to 200 kS/s.
3. Plot the power spectrum of v(t) on the ELVISmx Dynamic Signal Analyzer (DSA). Carefully adjust the panel controls to match the following settings:
   • Input Settings:
     1. Source Channel = AI0
     2. Voltage Range = +/-10V
   • FFT Settings:
     1. Frequency Span = 10000
     2. Resolution (lines) = 400
     3. Window = None
   • Averaging:
     1. Mode = RMS
     2. Weighting = Exponential
     3. Number of Averages = 5
   • Frequency Display:
     1. Units = Linear
     2. Mode = Peak
   • Scale Settings:
     1. Scale = Auto
   • Cursor Settings:
     1. Cursors On = enabled
     2. Cursor Select = C1
4. Measure the amplitude spectrum for n = 0 to 5 using Cursor 1; take the square root of the displayed cursor value “dVpk^2” to obtain the voltage amplitude. IMPORTANT: Position Cursor 2 between a pair of spectral lines to set its measured value to zero; the value displayed as dVpk^2 is the difference between the two cursors and you want Cursor 2 to serve as the zero reference.

Additional helpful tips:

• Use the cursor position buttons to make fine adjustments in the vicinity of a spectral line; these are the pair of gray diamonds at the bottom center of the DSA.
• Double-click the upper limit value on the horizontal frequency axis and select a lower value to zoom in on the lower-frequency spectral lines. Do not change the “Frequency Span” value for this purpose because this changes the measurement itself.

Further Exploration with NI myDAQ

The ELVISmx Digital Signal Analyzer (DSA) represents a sophisticated instrument that performs a wide variety of frequency-domain measurements. Experiment with the settings and discuss your findings:

• Averaging:
  1. Choose Mode = Peak Hold and note that the “Restart” button lower down becomes active. Also try Mode = None.
  2. Number of Averages: Try different values including 0.
• Frequency Display:
  1. Choose Units = dB; what advantage do you see in a logarithmic display compared to a linear display?

The “FFT Settings” control the Fast Fourier Transform computation that serves as the heart of the DSA. These critical settings must be carefully selected to obtain correct amplitude spectrum measurements of periodic signals. First learn how the DSA takes a measurement and then experiment with the settings in a moment.

The DSA repetitively captures a snapshot of the input signal with duration “Resolution (lines)” (R) divided by “Frequency Span” (f_span); this time-domain record appears below the frequency display. Take a moment to calculate this time duration from your current DSA FFT settings and confirm that the value does indeed match the upper limit of the time-domain plot.

When measuring a periodic signal the captured time-domain signal must contain an integer multiple of periods, consequently R/f_span divided by the signal period T must be an integer N. Since the periodic signal frequency f₀ is 1∕T , the frequency span may be readily calculated as

          f_span = Rf₀/N,

where f_span is the frequency span in Hz, R is the resolution in “lines” (sample points), f₀ is the fundamental frequency of the periodic input signal in Hz, and N is the number of periods captured. N = 10 cycles provides a good starting point for most measurements.

Now, return the DSA settings to match those of your earlier work in the NI myDAQ section of this problem. Calculate the value of N. Also calculate the values of N that result from choosing the other available values for resolution R (the DSA offers a total of five resolutions). Change the DSA FFT resolution to each of the other available values, and note the effects on the frequency spectrum display and on the time-domain display. In particular, note the degree to which the amplitude line spectral values change.

Return the resolution to R = 400 lines. Calculate the frequency span f_span for N = 10.5, i.e., for a time-domain record that contains ten periods with a half-period tacked onto the end. Enter this value into the DSA and note the degree to which the amplitude line spectral values change.