Of the many techniques currently in vogue for spectral estimation, the classical Fourier transform (FT) method is the most straightforward. The Fourier trans-form approach takes advantage of the fact that sinusoids contain energy at only one frequency. If a waveform can be broken down into a series of sines or co-sines of different frequencies, the amplitude of these sinusoids must be propor-tional to the frequency component contained in the waveform at those frequencies.
From Fourier series analysis, we know that any periodic waveform can be represented by a series of sinusoids that are at the same frequency as, or multi-ples of, the waveform frequency. This family of sinusoids can be expressed either as sines and cosines, each of appropriate amplitude, or as a single sine wave of appropriate amplitude and phase angle. Consider the case where sines and cosines are used to represent the frequency components: to find the appro-priate amplitude of these components it is only necessary to correlate (i.e., mul-tiply) the waveform with the sine and cosine family, and average (i.e., integrate) over the complete waveform (or one period if the waveform is periodic). Ex-pressed as an equation, this procedure becomes:
a(m)=1
T
∫
0Tx(t) cos(2πmfTt) dt (1)b(m)=1
T
∫
0Tx(t) sin(2πmfTt) dt (2)where T is the period or time length of the waveform, fT= 1/T, and m is set of integers, possibly infinite: m= 1, 2, 3, . . . , defining the family member. This gives rise to a family of sines and cosines having harmonically related frequen-cies, mfT.
In terms of the general transform discussed inChapter 2,the Fourier series analysis uses a probing function in which the family consists of harmonically
related sinusoids. The sines and cosines in this family have valid frequencies only at values of m/T, which is either the same frequency as the waveform (when m= 1) or higher multiples (when m > 1) that are termed harmonics.
Since this approach represents waveforms by harmonically related sinusoids, the approach is sometimes referred to as harmonic decomposition. For periodic functions, the Fourier transform and Fourier series constitute a bilateral trans-form: the Fourier transform can be applied to a waveform to get the sinusoidal components and the Fourier series sine and cosine components can be summed to reconstruct the original waveform:
x(t)= a(0)/2 +
∑
∞m=0
a(k) cos(2πmfTt)+
∑
∞m=0
b(k) sin (2πmfTt) (3) Note that for most real waveforms, the number of sine and cosine compo-nents that have significant amplitudes is limited, so that a finite, sometimes fairly short, summation can be quite accurate. Figure 3.3 shows the construction
FIGURE3.3 Two periodic functions and their approximations constructed from a limited series of sinusoids. Upper graphs: A square wave is approximated by a series of 3 and 6 sine waves. Lower graphs: A triangle wave is approximated by a series of 3 and 6 cosine waves.
of a square wave (upper graphs) and a triangle wave (lower graphs) using Eq.
(3) and a series consisting of only 3 (left side) or 6 (right side) sine waves. The reconstructions are fairly accurate even when using only 3 sine waves, particu-larly for the triangular wave.
Spectral information is usually presented as a frequency plot, a plot of sine and cosine amplitude vs. component number, or the equivalent frequency.
To convert from component number, m, to frequency, f, note that f= m/T, where T is the period of the fundamental. (In digitized signals, the sampling frequency can also be used to determine the spectral frequency). Rather than plot sine and cosine amplitudes, it is more intuitive to plot the amplitude and phase angle of a sinusoidal wave using the rectangular-to-polar transformation:
a cos(x)+ b sin(x) = C sin(x + Θ) (4)
where C= (a2+ b2)1/2andΘ = tan−1(b/a).
Figure 3.4 shows a periodic triangle wave (sometimes referred to as a sawtooth), and the resultant frequency plot of the magnitude of the first 10 components. Note that the magnitude of the sinusoidal component becomes quite small after the first 2 components. This explains why the triangle function can be so accurately represented by only 3 sine waves, as shown inFigure 3.3.
FIGURE3.4 A triangle or sawtooth wave (left) and the first 10 terms of its Fourier series (right). Note that the terms become quite small after the second term.
Symmetry
Some waveforms are symmetrical or anti-symmetrical about t= 0, so that one or the other of the components, a(k) or b(k) in Eq. (3), will be zero. Specifically, if the waveform has mirror symmetry about t= 0, that is, x(t) = x(−t), than mul-tiplications by a sine functions will be zero irrespective of the frequency, and this will cause all b(k) terms to be zeros. Such mirror symmetry functions are termed even functions. Similarly, if the function has anti-symmetry, x(t)= −x(t), a so-called odd function, then all multiplications with cosines of any frequency will be zero, causing all a(k) coefficients to be zero. Finally, functions that have half-wave symmetry will have no even coefficients, and both a(k) and b(k) will be zero for even m. These are functions where the second half of the period looks like the first half flipped left to right; i.e., x(t)= x(T − t). Functions having half-wave symmetry can also be either odd or even functions. These symmetries are useful for reducing the complexity of solving for the coefficients when such computations are done manually. Even when the Fourier transform is done on a computer (which is usually the case), these properties can be used to check the correctness of a program’s output. Table 3.1 summarizes these properties.
Discrete Time Fourier Analysis
The discrete-time Fourier series analysis is an extension of the continuous analy-sis procedure described above, but modified by two operations: sampling and windowing. The influence of sampling on the frequency spectra has been cov-ered inChapter 2.Briefly, the sampling process makes the spectra repetitive at frequencies mfT (m= 1,2,3, . . . ), and symmetrically reflected about these fre-quencies (seeFigure 2.9).Hence the discrete Fourier series of any waveform is theoretically infinite, but since it is periodic and symmetric about fs/2, all of the information is contained in the frequency range of 0 to fs/2 ( fs/2 is the Nyquist frequency). This follows from the sampling theorem and the fact that the origi-nal aorigi-nalog waveform must be bandlimited so that its highest frequency, fMAX, is <fs/2 if the digitized data is to be an accurate representation of the analog waveform.
The digitized waveform must necessarily be truncated at least to the length of the memory storage array, a process described as windowing. The windowing process can be thought of as multiplying the data by some window shape (see Figure 2.4).If the waveform is simply truncated and no further shaping is per-formed on the resultant digitized waveform (as is often the case), then the win-dow shape is rectangular by default. Other shapes can be imposed on the data by multiplying the digitized waveform by the desired shape. The influence of such windowing processes is described in a separate section below.
The equations for computing Fourier series analysis of digitized data are the same as for continuous data except the integration is replaced by summation.
Usually these equations are presented using complex variables notation so that both the sine and cosine terms can be represented by a single exponential term using Euler’s identity:
ejx= cos x + j sin x (5)
(Note mathematicians use i to represent
√
−1 while engineers use j; i is reserved for current.) Using complex notation, the equation for the discrete Fourier trans-form becomes:X(m)=
∑
N−1n=0
x(n)e(−j2πmn/N) (6)
where N is the total number of points and m indicates the family member, i.e., the harmonic number. This number must now be allowed to be both positive and negative when used in complex notation: m= −N/2, . . . , N/2–1. Note the similarity of Eq. (6) with Eq. (8) ofChapter 2,the general transform in discrete form. In Eq. (6), fm(n) is replaced by e−j2πmn/N. The inverse Fourier transform can be calculated as:
x(n)= 1 N
∑
N−1n=0
X(m) e−j2πnfmTs (7)
Applying the rectangular-to-polar transformation described in Eq. (4), it is also apparent*X(m)* gives the magnitude for the sinusoidal representation of the Fourier series while the angle of X(m) gives the phase angle for this repre-sentation, since X(m) can also be written as:
X(m)=
∑
N−1n=0x(n) cos(2πmn/N) − j∑
N−1n=0 x(n) sin(2πmn/N) (8) As mentioned above, for computational reasons, X(m) must be allowed to have both positive and negative values for m; negative values imply negative frequencies, but these are only a computational necessity and have no physical meaning. In some versions of the Fourier series equations shown above, Eq. (6)is multiplied by Ts(the sampling time) while Eq. (7) is divided by Tsso that the sampling interval is incorporated explicitly into the Fourier series coefficients.
Other methods of scaling these equations can be found in the literature.
The discrete Fourier transform produces a function of m. To convert this to frequency note that:
fm= mf1= m/TP= m/NTs= mfs/N (9)
where f1≡ fTis the fundamental frequency, Ts is the sample interval; fs is the sample frequency; N is the number of points in the waveform; and TP= NTs is the period of the waveform. Substituting m= fmTsinto Eq. (6), the equation for the discrete Fourier transform (Eq. (6)) can also be written as:
X(f )=
∑
N−1n=0x(n) e(−j2πnfmTs) (10)which may be more useful in manual calculations.
If the waveform of interest is truly periodic, then the approach described above produces an accurate spectrum of the waveform. In this case, such analy-sis should properly be termed Fourier series analyanaly-sis, but is usually termed Fourier transform analysis. This latter term more appropriately applies to aperi-odic or truncated waveforms. The algorithms used in all cases are the same, so the term Fourier transform is commonly applied to all spectral analyses based on decomposing a waveform into sinusoids.
Originally, the Fourier transform or Fourier series analysis was imple-mented by direct application of the above equations, usually using the complex formulation. Currently, the Fourier transform is implemented by a more compu-tationally efficient algorithm, the fast Fourier transform (FFT), that cuts the number of computations from N2to 2 log N, where N is the length of the digital data.
Aperiodic Functions
If the function is not periodic, it can still be accurately decomposed into sinu-soids if it is aperiodic; that is, it exists only for a well-defined period of time, and that time period is fully represented by the digitized waveform. The only difference is that, theoretically, the sinusoidal components can exist at all fre-quencies, not just multiple frequencies or harmonics. The analysis procedure is the same as for a periodic function, except that the frequencies obtained are really only samples along a continuous frequency spectrum. Figure 3.5 shows the frequency spectrum of a periodic triangle wave for three different periods.
Note that as the period gets longer, approaching an aperiodic function, the spec-tral shape does not change, but the points get closer together. This is reasonable
FIGURE 3.5 A periodic waveform having three different periods: 2, 2.5, and 8 sec. As the period gets longer, the shape of the frequency spectrum stays the same but the points get closer together.
since the space between the points is inversely related to the period (m/T ).* In the limit, as the period becomes infinite and the function becomes truly aperi-odic, the points become infinitely close and the curve becomes continuous. The analysis of waveforms that are not periodic and that cannot be completely repre-sented by the digitized data is described below.
*The trick of adding zeros to a waveform to make it appear to a have a longer period (and, therefore, more points in the frequency spectrum) is another example of zero padding.
Frequency Resolution
From the discrete Fourier series equation above (Eq. (6)), the number of points produced by the operation is N, the number of points in the data set. However, since the spectrum produced is symmetrical about the midpoint, N/2 (or fs/2 in frequency), only half the points contain unique information.* If the sampling time is Ts, then each point in the spectra represents a frequency increment of 1/(NTs). As a rough approximation, the frequency resolution of the spectra will be the same as the frequency spacing, 1/(NTs). In the next section we show that frequency resolution is also influenced by the type of windowing that is applied to the data.
As shown inFigure 3.5,frequency spacing of the spectrum produced by the Fourier transform can be decreased by increasing the length of the data, N.
Increasing the sample interval, Ts, should also improve the frequency resolution, but since that means a decrease in fs, the maximum frequency in the spectra, fs/2 is reduced limiting the spectral range. One simple way of increasing N even after the waveform has been sampled is to use zero padding, as was done in Figure 3.5. Zero padding is legitimate because the undigitized portion of the waveform is always assumed to be zero (whether true or not). Under this as-sumption, zero padding simply adds more of the unsampled waveform. The zero-padded waveform appears to have improved resolution because the fre-quency interval is smaller. In fact, zero padding does not enhance the underlying resolution of the transform since the number of points that actually provide information remains the same; however, zero padding does provide an interpo-lated transform with a smoother appearance. In addition, it may remove ambigu-ities encountered in practice when a narrowband signal has a center frequency that lies between the 1/NTsfrequency evaluation points (compare the upper two spectra in Figure 3.5). Finally, zero padding, by providing interpolation, can make it easier to estimate the frequency of peaks in the spectra.
Truncated Fourier Analysis: Data Windowing
More often, a waveform is neither periodic or aperiodic, but a segment of a much longer—possibly infinite—time series. Biomedical engineering examples are found in EEG and ECG analysis where the waveforms being analyzed con-tinue over the lifetime of the subject. Obviously, only a portion of such wave-forms can be represented in the finite memory of the computer, and some atten-tion must be paid to how the waveform is truncated. Often a segment is simply
*Recall that the Fourier transform contains magnitude and phase information. There are N/2 unique magnitude data points and N/2 unique phase data points, so the same number of actual data points is required to fully represent the data. Both magnitude and phase data are required to reconstruct the original time function, but we are often only interested in magnitude data for analysis.
cut out from the overall waveform; that is, a portion of the waveform is trun-cated and stored, without modification, in the computer. This is equivalent to the application of a rectangular window to the overall waveform, and the analysis is restricted to the windowed portion of the waveform. The window function for a rectangular window is simply 1.0 over the length of the window, and 0.0 else-where, (Figure 3.6, left side). Windowing has some similarities to the sampling process described previously and has well-defined consequences on the resultant frequency spectrum. Window shapes other than rectangular are possible simply by multiplying the waveform by the desired shape (sometimes these shapes are referred to as tapering functions). Again, points outside the window are assumed to be zero even if it is not true.
When a data set is windowed, which is essential if the data set is larger than the memory storage, then the frequency characteristics of the window be-come part of the spectral result. In this regard, all windows produce artifact. An idea of the artifact produced by a given window can be obtained by taking the Fourier transform of the window itself. Figure 3.6 shows a rectangular window on the left side and its spectrum on the right. Again, the absence of a window function is, by default, a rectangular window. The rectangular window, and in fact all windows, produces two types of artifact. The actual spectrum is widened by an artifact termed the mainlobe, and additional peaks are generated termed
FIGURE3.6 The time function of a rectangular window (left) and its frequency characteristics (right).
the sidelobes. Most alternatives to the rectangular window reduce the sidelobes (they decay away more quickly than those ofFigure 3.6),but at the cost of wider mainlobes. Figures 3.7 and3.8show the shape and frequency spectra produced by two popular windows: the triangular window and the raised cosine or Ham-ming window. The algorithms for these windows are straightforward:
Triangular window:
for odd n:
w(k)=再2k/(n2(n− k − 1)/(n + 1)− 1)
1≤ k ≤ (n + 1)/2
(n+ 1)/2 ≤ k ≤ n (11)
for even n:
w(k)=再(2k2(n− 1)/n− k + 1)/n 1(n/2)≤ k ≤ n/2+ 1 ≤ k ≤ n (12) Hamming window:
w(k+ 1) = 0.54 − 0.46(2πk/(n − 1))k = 0, 1, . . . , n − 1 (13)
FIGURE3.7 The triangular window in the time domain (left) and its spectral char-acteristic (right). The sidelobes diminish faster than those of the rectangular win-dow (Figure 3.6), but the mainlobe is wider.
FIGURE3.8 The Hamming window in the time domain (left) and its spectral char-acteristic (right).
These and several others are easily implemented in MATLAB, especially with the Signal Processing Toolbox as described in the next section. A MATLAB routine is also described to plot the spectral characteristics of these and other windows. Selecting the appropriate window, like so many other aspects of signal analysis, depends on what spectral features are of interest. If the task is to resolve two narrowband signals closely spaced in frequency, then a window with the narrowest mainlobe (the rectangular window) is preferred. If there is a strong and a weak signal spaced a moderate distance apart, then a window with rapidly decaying sidelobes is preferred to prevent the sidelobes of the strong signal from overpowering the weak signal. If there are two moderate strength signals, one close and the other more distant from a weak signal, then a compro-mise window with a moderately narrow mainlobe and a moderate decay in side-lobes could be the best choice. Often the most appropriate window is selected by trial and error.
Power Spectrum
The power spectrum is commonly defined as the Fourier transform of the auto-correlation function. In continuous and discrete notation, the power spectrum equation becomes:
PS(f )=
∫
where rxx(n) is the autocorrelation function described in Chapter 2. Since the autocorrelation function has odd symmetry, the sine terms, b(k) will all be zero (seeTable 3.1)and Eq. (14) can be simplified to include only real cosine terms.
PS(f )=
∫
These equations in continuous and discrete form are sometimes referred to as the cosine transform. This approach to evaluating the power spectrum has lost favor to the so-called direct approach, given by Eq. (18) below, primarily because of the efficiency of the fast Fourier transform. However, a variation of this approach is used in certain time–frequency methods described in Chapter 6.One of the problems compares the power spectrum obtained using the direct approach of Eq. (18) with the traditional method represented by Eq. (14).
The direct approach is motivated by the fact that the energy contained in an analog signal, x(t), is related to the magnitude of the signal squared, inte-grated over time:
E=
∫
∞
−∞
*x(t)*2dt (16)
By an extension of Parseval’s theorem it is easy to show that:
∫
−∞∞ *x(t)*2dt=∫
−∞∞ *X(f)*2df (17)Hence*X( f )*2equals the energy density function over frequency, also re-ferred to as the energy spectral density, the power spectral density, or simply the power spectrum. In the direct approach, the power spectrum is calculated as the magnitude squared of the Fourier transform of the waveform of interest:
PS(f )= *X(f)*2 (18)
Power spectral analysis is commonly applied to truncated data, particu-larly when the data contains some noise, since phase information is less useful in such situations.
While the power spectrum can be evaluated by applying the FFT to the entire waveform, averaging is often used, particularly when the available wave-form is only a sample of a longer signal. In such very common situations, power spectrum evaluation is necessarily an estimation process, and averaging im-proves the statistical properties of the result. When the power spectrum is based on a direct application of the Fourier transform followed by averaging, it is com-monly referred to as an average periodogram. As with the Fourier transform, evaluation of power spectra involves necessary trade-offs to produce statistically
While the power spectrum can be evaluated by applying the FFT to the entire waveform, averaging is often used, particularly when the available wave-form is only a sample of a longer signal. In such very common situations, power spectrum evaluation is necessarily an estimation process, and averaging im-proves the statistical properties of the result. When the power spectrum is based on a direct application of the Fourier transform followed by averaging, it is com-monly referred to as an average periodogram. As with the Fourier transform, evaluation of power spectra involves necessary trade-offs to produce statistically