Originally posted Nov 2, 2015
How to discard information and improve your measurements
If your primary focus is RF/microwave analysis, you may not be familiar with “windows” or “window functions,” and they may not be a factor in your explicit measurement choices. However, it’s worth knowing a little about them for at least two reasons: you may already be using them, and they can help you makebetter measurements in demanding situations.
Windows have been an essential feature of the fast Fourier transform (FFT) architecture of low-frequency analyzers for many years. FFT processing has also been added to many high-frequency analyzers as a way to implement narrow resolution bandwidths (RBWs) while optimizing measurement speed. Finally, FFT processing is central to vector signal analyzers (VSAs) and OFDM demodulation in particular.
FFTs calculate a spectrum from a block of samples called a time record, and the FFT algorithm assumes that the time record repeats endlessly. That assumption is valid for signals that are periodic over the selected time record, but it causes discontinuity errors for signals that are not. In the FFT spectrum results, the errors create a spreading of the spectral energy called leakage.
The solution is to somehow force the signal to be periodic within the time record, and the most common approach is to multiply the time record by a weighting function that reduces amplitude to zero at both ends of the time record, as shown below.
In this example of a non-repeating sine wave, the FFT algorithm’s assumption that signals repeat for each time record produces the erroneous signal in the second waveform. The window or weighting function removes the discontinuities before the FFT calculates the spectrum.
As an engineer, you’d expect tradeoffs from this weighting process, which effectively discards some samples and down-weights others. That is indeed the case and, among other things, the windowing widens the RBW. It also creates sidelobes of varying amplitude and frequency spacing, depending on the window shape.
The good news is that window shapes can be chosen to optimize the tradeoffs for specific measurements, such as prioritizing frequency resolution, amplitude accuracy or sidelobe level.
I’ll discuss examples of those tradeoffs in a future post, but first I’d like to show what’s possible in the best-case, where the signal is periodic in the time record and the uniform window—equivalent to no windowing—can be used.
Two gated spectrum measurements are made of an off-air capture of an 802.11n signal in the 5 GHz band. The gate is set to match a portion of the training sequence, which is periodic or self-windowing. The uniform window of the 89600 VSA software can be used in this case, providing enough frequency resolution in the bottom trace to measure individual OFDM subcarriers.
In this measurement, gated sweep in the 89600 VSA software has been configured to align with a portion of the training sequence which is self-windowing. The selected uniform window is actually no window at all, and no signal samples are discarded or down-weighted. In this special case no tradeoffs are needed between accuracy, frequency resolution and dynamic range.
As an aside, this training sequence includes every second subcarrier, transmitted at equal amplitude. The peaks describe the ragged frequency response that receivers have to deal with in the real world.
Vector signal analyzers use FFTs for spectrum analysis all the time, but modern signal analyzers such as Keysight’s X-Series automatically choose between FFTs and swept digital filters as needed. In a future post or two I’ll discuss how to optimize FFT analysis and select windows to extract maximum information and improve measurement speed in swept measurements.