# Bob & Ken's Spectrum Inversion Trick

I learned a clever trick from Bob Smith and Ken Wetzel, each of whom independently discovered it. The problem is this: how do you measure the distortion of an FM signal generator? The difficulty is that any FM detector you use will have its own distortion. How do you separate the distortion of the generator from that of the detector?

I never imagined that it might be possible. My approach was to build the most linear FM detector I could, usually a pulse counter, and then assume it was perfectly linear. It can't be, of course, but careful design and attention to detail will minimize nonlinearities. However, I had no way to measure whatever residual distortion remained.

Bob and Ken's trick lets you independently characterize even-order distortion products for any generator and detector. It relies on waveform asymmetry. Any nonlinearity that follows a power law, where the power is an even integer, causes the shape of positive and negative waveform excursions to differ.

This shows a sine wave with 25% second-harmonic distortion. When such a signal is frequency modulated, the instantaneous RF frequency traces out a curve like that shown. If you were to somehow invert the RF spectrum, the detected waveform would invert. The key insight is that inversion flips by 180° not only the phase of the fundamental, but also that of even-order distortion products. Generator distortion that added to detector distortion before inversion will subtract from it after inversion. Measured distortion values will differ for normal and inverted spectra, and a little algebra will separate the generator and detector contributions.

There's one more benefit. If the generator has adjustable distortion compensation, vary it until the distortion products for normal and inverted spectra are equal. The resulting value is the detector distortion, and you've just nulled the generator distortion. Since the dominant distortion product for a varactor frequency modulator is the second, in practice this means that you can null the generator's second harmonic. What's surprising is that the detector's second harmonic need not be zero to accomplish this.

Let g be generator distortion, d the detector distortion, and m1 and m2 the measured distortion for normal and inverted spectra, all linear amplitudes of the second-order product. Then when g < d,

d + g = m1
d - g = m2

d = (m1 + m2) / 2
g = (m1 - m2) / 2

When m1 = m2, d = m1 and g = 0. If you can measure the difference between m1 and m2 to within 1 dB, g will be at least 25 dB below d. At this tolerance, the second-harmonic contribution to generator THD will be less than .005% when the detector distortion is 61 dB down.

To invert the signal spectrum, you can use a mixer and a second signal generator. In one case set it below the signal frequency and in the other case above. For example, to characterize a signal generator at 98 MHz, set an FM tuner at 108 MHz and the second generator at 10 MHz. The mixer sum product at 108 MHz will be noninverted. Then set the second generator at 206 MHz to invert the 108 MHz difference product. The frequencies of the generator under test and the tuner remain constant so their distortion levels don't change. A +7 dBm LO diode ring mixer with a 1 dB compression point of +1 dBm will not add distortion at the usual 65 dBf generator test level of −55 dBm.

Instead of a mixer and second generator, I use my Marconi 2305 modulation meter. This instrument has a wideband, low-distortion, charge-pump FM detector (a specific kind of pulse counter). Because it does not use an input bandpass filter, I can tune to the signal frequency or to the image 3 MHz below to obtain a signal at the 1.5 MHz IF. The spectrum inverts when I retune.

Bob and Ken's trick won't help with odd-order distortion. Such waveforms, like this sine wave with a 10% third harmonic, are always symmetrical. But harmonic levels tend to decrease with product order, and higher-order products normally drop much more rapidly than the fundamental and second harmonic as the signal amplitude decreases. For example, the baseband fundamental of a stereo signal is about 6 dB below its monophonic level. This can drop the second harmonic of a high-quality signal generator near the noise floor and higher-order products below it. While odd-order distortion products can't be entirely neglected, the second harmonic usually dominates, and you can minimize or eliminate it with Bob & Ken's spectrum inversion trick.

December 7, 201288–108 MHz