# Spectral Estimation Techniques

Common spectral estimation techniques appear to be FFT based, with there being many different FFT approaches (for example Welch's Method or STFT). However Wikipedia lists many other possible techniques, such as autoregression and least squares.

Most RF spectrum analysers appear to use the FFT approach. Are these other techniques, like autoregression or least squares, widely used? If so why? If not, why not?

It’s very application dependent. Spectrum analyzers are typically performing real-time estimates, so using FFT methods is more efficient than modern spectral estimation methods, since most modern estimates need to invert a correlation matrix, which is $$\mathcal{O}(N^{2})$$ if you use a toeplitz estimate and can use Levinson recursion.

One place where spectral estimation found a niche was in SAR imaging. You could form the image, transmit it, then post-process it via spectral estimation. It’s a little bit easier to do spectral analysis with spatial signals (eg DOA/Beamforming) because you often don’t have as many samples as you would in a temporal signal.

There are two main approaches to spectral analysis: filterbank and model based approaches. Classical Fourier methods (that you refer to as FFT based methods), are based on non-adaptive filterbanks. Modern filterbank methods are based on adaptive filterbanks, ie RFB, Capon, etc. Then there are also finite-dimensional models like AR/LP, MUSIC, etc. Some of these techniques also have the ability to be computed via FFT, although a bit more indirectly. The idea behind these modern methods is that they have better statistical properties as most of them are both asymptotically unbiased and consistent, unlike classical Fourier methods.

• Thanks! Is the difference between filter-bank and model-based approaches the same as the difference between parametric vs non-parametric approaches? Commented Apr 22 at 18:07
• @DanielSomervilleRoberts yes, non-parametric = filterbank. There are two ways of estimating a continuous spectrum, using a finite number of bandpass filters or a finite dimensional model. Commented Apr 22 at 18:17
• This is an incredible answer! Can you perhaps share a link to an good overview of spectral estimation methods? Commented Apr 23 at 2:11
• @YairM thanks! For technical details, the Stoica and Moses book is a good book. It covers a wide variety of different techniques with fairly straightforward derivations while also getting into the nitty gritty details. Commented Apr 23 at 2:42

Depends on the application.

FFT-based methods are computationally efficient, so they are well-suited to real-time spectral analysis. Many digital signal processors and specialized hardware are optimized to perform FFT operations, which makes FFT based estimators a natural choice for embedded systems in spectrum analyzers.

Model-based and parametric estimation through AR and least squares methods are more common in research and in applications requiring detailed analysis of a smaller set of data or where model-based approaches provide additional benefits (e.g., in echo cancellation, speech analysis, seismic data, financial data, etc).

In essence, when the model is known and analysis can be done offline, high-resolution methods are better suited than FFT based approaches which are non-parametric (simple, and applicable to a wide range of signals) and computationally efficient and well suited for real-time processing (such as RF spectrum analyzers).

• Thanks! So is it the case that FFT based approaches are generally less accurate / precise than model-based / parametric approaches? Commented Apr 22 at 18:04
• Parametric and model based approaches are more precise if the parameters of your model are well-defined.
– Jdip
Commented Apr 22 at 18:32

One important application where FFT is difficult to apply is spectrums with wide dynamic range.

Sensitive laboratory instruments (such as EMI receivers) often perform analog downconversion and measure only a narrow band at a time. This way they can handle 100 dB or even higher power differences between frequencies.

If you try to directly sample a signal with wide dynamic range, the larger signal will limit the amplification you can use and the smaller signal gets lost in quantization noise. With non-ideal ADC, the larger signal will also be slightly distorted and cause harmonic noise in FFT results of other frequencies, further disguising small signals.

These analog-domain solutions to the dynamic range problem may not be exactly what you are asking about, as they involve hardware changes. But even using the same data, parametric methods often have better noise reduction than FFT because the model includes assumptions about the signal.