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The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. 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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
purposes, sampling is modeled as a product between s(t) and a Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
function. The spectrum of a product is the convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
between S(f) and another function, which inevitably creates the new frequency components. But the term 'leakage' usually refers to the effect of ''windowing'', which is the product of s(t) with a different kind of function, the window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
. Window functions happen to have finite duration, but that is not necessary to create leakage. Multiplication by a time-variant function is sufficient.
Spectral analysis
TheFourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of the function is zero, except at frequency ±''ω''. However, many other functions and waveforms do not have convenient closed-form transforms. Alternatively, one might be interested in their spectral content only during a certain time period. In either case, the Fourier transform (or a similar transform) can be applied on one or more finite intervals of the waveform. In general, the transform is applied to the product of the waveform and a window function. Any window (including rectangular) affects the spectral estimate computed by this method.
The effects are most easily characterized by their effect on a sinusoidal s(t) function, whose unwindowed Fourier transform is zero for all but one frequency. The customary frequency of choice is 0 Hz, because the windowed Fourier transform is simply the Fourier transform of the window function itself (see ):
:
When both sampling and windowing are applied to s(t), in either order, the leakage caused by windowing is a relatively localized spreading of frequency components, with often a blurring effect, whereas the aliasing caused by sampling is a periodic repetition of the entire blurred spectrum.
Choice of window function
Windowing of a simple waveform like causes its Fourier transform to develop non-zero values (commonly called spectral leakage) at frequencies other than ''ω''. The leakage tends to be worst (highest) near ''ω'' and least at frequencies farthest from ''ω''. If the waveform under analysis comprises two sinusoids of different frequencies, leakage can interfere with our ability to distinguish them spectrally. Possible types of interference are often broken down into two opposing classes as follows: If the component frequencies are dissimilar and one component is weaker, then leakage from the stronger component can obscure the weaker one's presence. 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 ''highdynamic range
Dynamic range (abbreviated DR, DNR, or DYR) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base ...
''. 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-resolution, low-dynamic-range windows such as the rectangular window also have the property of high ''sensitivity'', which is the ability to reveal relatively weak sinusoids in the presence of additive random noise. That is because the noise produces a stronger response with high-dynamic-range windows than with high-resolution windows.
At the other extreme of the range of window types are windows with high dynamic range but low resolution and sensitivity. 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 between the extremes are moderate windows, such as Hann and Hamming Hamming may refer to:
* Richard Hamming (1915–1998), American mathematician
* Hamming(7,4), in coding theory, a linear error-correcting code
* Overacting, or acting in an exaggerated way
See also
* Hamming code, error correction in telecom ...
. They are commonly used in ''narrowband applications'', such as the spectrum of a telephone channel.
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''). That trade-off occurs when the window function is chosen.
Discrete-time signals
When the input waveform is time-sampled, instead of continuous, the analysis is usually done by applying a window function and then adiscrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
(DFT). But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
(DTFT) spectrum. Figure 2, row 3 shows a DTFT for a rectangularly-windowed sinusoid. The actual frequency of the sinusoid is indicated as "13" on the horizontal axis. Everything else is leakage, exaggerated by the use of a logarithmic presentation. The unit of frequency is "DFT bins"; that is, the integer values on the frequency axis correspond to the frequencies sampled by the DFT. So the figure depicts a case where the actual frequency of the sinusoid coincides with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample. In row 4, it misses the maximum value by ½ 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.
Noise bandwidth
The concepts of resolution and dynamic range tend to be somewhat subjective, depending on what the user is actually trying to do. But they also tend to be highly correlated with the total leakage, which is quantifiable. It is usually expressed as an equivalent bandwidth, B. It can be thought of as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B. The more the leakage, the greater the bandwidth. It is sometimes called ''noise equivalent bandwidth'' or ''equivalent noise bandwidth'', because it is proportional to the average power that will be registered by each DFT bin when the input signal contains a random noise component (or is just random noise). A graph of thepower spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
, averaged over time, typically reveals a flat '' noise floor'', caused by this effect. The height of the noise floor is proportional to B. So two different window functions can produce different noise floors, as seen in figures 1 and 3.
Processing gain and losses
Insignal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, operations are chosen to improve some aspect of quality of a signal by exploiting the differences between the signal and the corrupting influences. When the signal is a sinusoid corrupted by additive random noise, spectral analysis distributes the signal and noise components differently, often making it easier to detect the signal's presence or measure certain characteristics, such as amplitude and frequency. Effectively, the signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in de ...
(SNR) is improved by distributing the noise uniformly, while concentrating most of the sinusoid's energy around one frequency. ''Processing gain'' is a term often used to describe an SNR improvement. The processing gain of spectral analysis depends on the window function, both its noise bandwidth (B) and its potential scalloping loss. These effects partially offset, because windows with the least scalloping naturally have the most leakage.
Figure 3 depicts the effects of three different window functions on the same data set, comprising two equal strength sinusoids in additive noise. The frequencies of the sinusoids are chosen such that one encounters no scalloping and the other encounters maximum scalloping. Both sinusoids suffer less SNR loss under the Hann window than under the Blackman-Harris window. In general (as mentioned earlier), this is a deterrent to using high-dynamic-range windows in low-dynamic-range applications.
Symmetry
The formulas provided at produce discrete sequences, as if a continuous window function has been "sampled". (See an example at Kaiser window.) Window sequences for spectral analysis are either ''symmetric'' or 1-sample short of symmetric (called ''periodic'', ''DFT-even'', or ''DFT-symmetric''). For instance, a true symmetric sequence, with its maximum at a single center-point, is generated by theMATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
function hann(9,'symmetric'). Deleting the last sample produces a sequence identical to hann(8,'periodic'). Similarly, the sequence hann(8,'symmetric') has two equal center-points.
Some functions have one or two zero-valued end-points, which are unnecessary in most applications. Deleting a zero-valued end-point has no effect on its DTFT (spectral leakage). But the function designed for + 1 or + 2 samples, in anticipation of deleting one or both end points, typically has a slightly narrower main lobe, slightly higher sidelobes, and a slightly smaller noise-bandwidth.
DFT-symmetry
The predecessor of the DFT is thefinite Fourier transform __NOTOC__
In mathematics the finite Fourier transform may refer to either
*another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the ''finite Fourier transform'' as a "co ...
, and window functions were "always an odd number of points and exhibit even symmetry about the origin". In that case, the DTFT is entirely real-valued. When the same sequence is shifted into a ''DFT data window'', the DTFT becomes complex-valued except at frequencies spaced at regular intervals of Thus, when sampled by an -length DFT, the samples (called DFT coefficients) are still real-valued. An approximation is to truncate the +1-length sequence (effectively ), and compute an -length DFT. The DTFT (spectral leakage) is slightly affected, but the samples remain real-valued.
The terms ''DFT-even'' and ''periodic'' refer to the idea that if the truncated sequence were repeated periodically, it would be even-symmetric about and its DTFT would be entirely real-valued. But the actual DTFT is generally complex-valued, except for the DFT coefficients. Spectral plots like those at , are produced by sampling the DTFT at much smaller intervals than and displaying only the magnitude component of the complex numbers.
Periodic summation
An exact method to sample the DTFT of an +1-length sequence at intervals of is described at . Essentially,