Spectral leakage

Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which changes the relative magnitudes and/or angles (phase) of the non-zero values of S(f).

Any other type of operation creates new frequency components that may be referred to as spectral leakage in the broadest sense.

Sampling, for instance, produces leakage, which we call aliases of the original spectral component.

For Fourier transform purposes, sampling is modeled as a product between s(t) and a Dirac comb function.

The spectrum of a product is the convolution between S(f) and another function, which inevitably creates the new frequency components.

Windowing of a simple waveform like cos(ωt) causes its Fourier transform to develop non-zero values (commonly called spectral leakage) at frequencies other than ω.

If the waveform under analysis comprises two sinusoids of different frequencies, leakage can interfere with our ability to distinguish them spectrally.

But if the frequencies are too similar, leakage can render them unresolvable even when the sinusoids are of equal strength.

Windows that are effective against the first type of interference, namely where components have dissimilar frequencies and amplitudes, are called high dynamic range.

Conversely, windows that can distinguish components with similar frequencies and amplitudes are called high resolution.

The rectangular window is an example of a window that is high resolution but low dynamic range, meaning it is good for distinguishing components of similar amplitude even when the frequencies are also close, but poor at distinguishing components of different amplitude even when the frequencies are far away.

High-dynamic-range windows are most often justified in wideband applications, where the spectrum being analyzed is expected to contain many different components of various amplitudes.

In summary, spectral analysis involves a trade-off between resolving comparable strength components with similar frequencies (high resolution / sensitivity) and resolving disparate strength components with dissimilar frequencies (high dynamic range).

[1]: p.90 When the input waveform is time-sampled, instead of continuous, the analysis is usually done by applying a window function and then a discrete Fourier transform (DFT).

But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum.

In row 4, it misses the maximum value by 1⁄2 bin, and the resultant measurement error is referred to as scalloping loss (inspired by the shape of the peak).

For a known frequency, such as a musical note or a sinusoidal test signal, matching the frequency to a DFT bin can be prearranged by choices of a sampling rate and a window length that results in an integer number of cycles within the window.

The concepts of resolution and dynamic range tend to be somewhat subjective, depending on what the user is actually trying to do.

It can be thought of as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B.

A graph of the power spectrum, averaged over time, typically reveals a flat noise floor, caused by this effect.

Effectively, the signal-to-noise ratio (SNR) is improved by distributing the noise uniformly, while concentrating most of the sinusoid's energy around one frequency.

The processing gain of spectral analysis depends on the window function, both its noise bandwidth (B) and its potential scalloping loss.

Figure 3 depicts the effects of three different window functions on the same data set, comprising two equal strength sinusoids in additive noise.

For instance, a true symmetric sequence, with its maximum at a single center-point, is generated by the MATLAB function hann(9,'symmetric').

[7] The predecessor of the DFT is the finite Fourier transform, and window functions were "always an odd number of points and exhibit even symmetry about the origin".

Spectral plots like those at § Examples of window functions, are produced by sampling the DTFT at much smaller intervals than

[14][2]: p.62 [1]: p.85  In those applications, DFT-symmetric windows (even or odd length) from the Cosine-sum family are preferred, because most of their DFT coefficients are zero-valued, making the convolution very efficient.

It is relative to the maximum possible response, which occurs when the signal frequency is an integer number of bins.

Figure 1: Comparison of two window functions in terms of their effects on equal-strength sinusoids with additive noise. The sinusoid at bin −20 suffers no scalloping and the one at bin +20.5 exhibits worst-case scalloping. The rectangular window produces the most scalloping but also narrower peaks and lower noise-floor. A third sinusoid with amplitude −16 dB would be noticeable in the upper spectrum, but not in the lower spectrum.
Figure 2: Windowing a sinusoid causes spectral leakage, even if the sinusoid has an integer number of cycles within a rectangular window. The leakage is evident in the 2nd row, blue trace. It is the same amount as the red trace, which represents a slightly higher frequency that does not have an integer number of cycles. When the sinusoid is sampled and windowed, its discrete-time Fourier transform also exhibits the same leakage pattern (rows 3 and 4). But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue sinusoid DTFT (3rd row of plots, right-hand side), those samples are the outputs of the discrete Fourier transform (DFT). The red sinusoid DTFT (4th row) has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed.
Figure 3: This figure compares the processing losses of three window functions for sinusoidal inputs, with both minimum and maximum scalloping loss.
Figure 4: Two different ways to generate an 8-point Gaussian window sequence ( σ = 0.4) for spectral analysis applications. MATLAB calls them "symmetric" and "periodic". The latter is also historically called DFT-even .
Figure 5: Spectral leakage characteristics of the functions in Figure 4
Comparison of spectral leakage of several window functions