Stanford Research Systems SR865A, Operation Manual

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The FFT Display 161
SR865A DSP Lock-in Amplifier
Appendix B
The FFT Display
The FFT display takes a time varying signal, like you would see on an oscilloscope trace,
and computes its frequency spectrum. In the SR865A, this signal is either the output of
the input amplifier (Raw ADC) or the output of the PSD (Post Mixer or Post Filter).
Fourier's basic theorem states that any waveform in the time domain can be represented
by the weighted sum of pure sine waves of all frequencies. If the signal in the time
domain (as viewed on an oscilloscope) is periodic, then its spectrum is probably
dominated by a single frequency component. What the FFT does is represent the time
domain signal by its component frequencies.
Why Look at the Spectrum?
A lock-in amplifier measures a single frequency component of the input signal at f
ref
.
With time constant filtering, all other frequencies are ignored and filtered away.
Sometimes it can be useful to see what’s going on at other frequencies, even if only with
a broad overview. This can help identify sources of interference or noise. Other times it
may be useful for estimating the signal at other frequencies without making multiple
measurements, each at different reference frequencies and time constants. Take harmonic
distortion. Instead of making separate measurements at f
ref
, 2×f
ref
, 3×f
ref
, etc. the FFT can
display the harmonic frequencies and amplitudes with amazing clarity in a single
measurement. Another example is noise analysis. In an FFT, the noise as a function of
frequency is displayed.
The FFT Analyzer
The lock-in amplifier digitizes the input signal and multiplies by f
ref
in a phase sensitive
detector yielding a very narrow band measurement at f
ref
.
An FFT analyzer works in an entirely different way. A time record of samples (1024
samples in the SR865A) is mathematically transformed into a frequency spectrum using
an algorithm known as the Fast Fourier Transform or FFT. The FFT is simply a clever set
of operations which implements Fourier's basic theorem. The resulting spectrum shows
the frequency components of the input signal.
Now here's the interesting part. The original digital time record comes from discrete
samples taken at the sampling rate. The corresponding FFT yields a spectrum with
discrete frequency samples. In fact, the spectrum has half as many frequency points as
there are time points. (Remember Nyquist's theorem). Suppose that you take 1024
samples at 256 kHz. It takes 4 ms to take this time record. The FFT of this record yields
512 frequency points, but over what frequency range? The highest frequency will be
determined by the period of 2 time samples or 128 kHz. The lowest frequency is just the
period of the entire record or 1/(4 ms) or 250 Hz. Everything below 250 Hz is considered