*Excel’s charting function adds nonharmonic frequency components to a cosine wave.*

In Part 1 and Part 2 of this series, we looked at intermodulation — what it is and how to apply it to frequency-translation tasks. We concluded Part 2 with an image like **Figure 1**. We had begun with a 1,200-kHz carrier with two sidelobe peaks at ±50-kHz offsets (shown as a dotted-blue trace). We then mixed this amplitude-modulated carrier with a 900-kHz sine wave and ran the Microsoft Excel fast Fourier transform (FFT) to analyze the result. We see in Figure 1 that we have eliminated our original signal, replacing it with two versions, upconverted and downconverted by 900 kHz (red trace). We noted that these conversions preserve the ±50-kHz offset sidelobes.

**Can I use the FFT template we developed in a recent ****series**** on FFT to experiment with frequency translation — perhaps by choosing more realistic ****±5-kHz sidelobes?**

Yes, but with one key change, which I can illustrate with a specific example. We can start with our 1,200-kHz carrier and amplitude-modulate it using a 5-kHz baseband signal instead of a 50-kHz one, in accordance with this equation:

where *f _{c}* = 1,200 kHz,

*f*= 5 kHz, and

_{m}*A*= 0.75.

_{m}Then, we can mix it with a 500-kHz sinusoid (*f _{mix}*) to obtain an upconverted 1,700-kHz signal—the highest AM radio frequency band in the U.S. — and a corresponding downconverted signal at 700 kHz. Then, we can analyze the resulting translated signal using the FFT.

**OK, so the first step would be to choose a sampling frequency. It seems a 2,000-kHz Nyquist frequency would suffice, implying a 4,000-kHz sampling frequency, right?**

Right, and here is the problem with our earlier template. It used a 512-point FFT. For a 4,000-kHz sampling frequency (for a 250-ns sampling interval), a 512-point FFT implies an FFT frequency-bin width of 7.81 kHz (4,000 kHz/512), so we clearly have insufficient frequency resolution for examining a 5-kHz offset.

**What are our options?**

You can improve the frequency resolution in two ways: by decreasing the sampling frequency (and hence increasing the sampling interval) or by increasing the number of points in the FFT. We have already suggested a sampling frequency that pushes the Nyquist frequency down close to our 1,700-kHz signal of interest. Our other option is to increase the number of points in the FFT. If we push the point count to 4,096 — the maximum that Excel supports — our frequency-bin width equals 4,000 kHz/4,096, or 0.977 kHz, a figure that should let us at least confirm the presence of the 5-kHz sidelobes. **Figure 2** shows some changes in red to the original FFT template necessary for this new calculation.

**Figure 3** shows the resulting FFT. Note that the sidelobes are not distinguishable in the gray-tinted plot from zero to the Nyquist frequency, but they are visible in the inset, which magnifies the region of the plot near the upconverted signal.

**We’ve covered the benefits of intermodulation for frequency translation, but what about intermodulation distortion?**

Let me provide an example. **Figure 4** shows our 1.7-MHz sinusoidal carrier in blue. Sampling occurs every 0.25 µs, with the sample points highlighted in yellow. Now, if we ask Excel to chart these sample points with the “smoothed line” function activated, we get the red trace.

To find out more about this red trace, I sampled it 60 times and ran an FFT on the resulting dataset. You can see the FFT in **Figure 5**. In addition to our 1.7-MH carrier, there is a significant peak at 2.3 MHz. We have fed a cosine wave into the black box of Excel’s “Smoothed line” function, and it generated a nonharmonic frequency component (we can attribute some of the noise to my carelessness in interpolating the 60 values for the red trace’s dataset — I am part of the black box in this example).

This process of applying sinusoidal signals to a black box and getting nonharmonic intermodulation products in return is common for both active and passive systems. We’ll take a closer look at this intermodulation distortion (IMD) and passive intermodulation (PIM) distortion in the next and final entry in this series.