Usual and unusual concepts
2.8 TIME AND SPECTRAL DOMAINS, HELICAL SPECTRA
Origin Signal vector
S
Added bivariate Gaussian noise with standard deviation F
Vector to signal + noise
sample r S
r Rayleigh root mean square = /2 Fxy
Median Fxy
dr
0 0.1 0.2 0.3 0.4 0.5
2 4 6 8 10
r S=0
S=1 S=2 S=3 S=4 S=5S=5 S=6 S=7
Rectangular pulse ' 1 for &J/2 < t < J/2 ' 0 otherwise
Figure 2.24 Bivariate Gaussian noise added to a signal vector.
Figure 2.25 Ricean distribution for various values of S.
(2.24) Figure 2.25 shows the Ricean distribution for F = 1 for various values of S. When S = 0, this is a Rayleigh distribution, which is used to determine the probability of detection for echoes that do not fade.
Spectrum '
I
%J
2
&J 2
1 e&j 2Bft dt
' sin BfJ
B fJ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1 -0.5
0.5 1 -2
-1
1 2
0 0.5 1
-2
-1
1
2
-1 -0.5
0.5 1
-0.2 0 0.5 1
-2
-1
1
2 -2
-1
1
2
Rectangular pulse waveform sin BfJ spectrum BfJ
Combined time waveform and spectrum Time
Frequency
J = 1 second Frequency
Time
Frequency
Time
Spectrum of the pulse delayed by t1 ' sin BfJ
BfJ e&j 2Bf t1
(2.25)
Figure 2.26 The assembly of a time waveform, width 1 s, and spectrum onto a single diagram.
(2.26) The spectrum for the pulse with a constant height of 1 (unity) along its length, centered on zero is
Figure 2.26 shows the combination of a time waveform and its spectrum into a single diagram. For an example the familiar rectangular pulse of width T and its sin x/x spectrum have been chosen.
Both the time waveform and its spectrum are real and can be easily represented on a flat piece of paper. The orientation and axes have been chosen such that:
• The common reference vector is along the real axis in time and frequency. This common direction is up as in polyphase vector diagrams;
• The diagrams represent the standard two-dimensional real form when one is looking from the front and the imaginary form when looking from above.
As the time pulse moves away from being centered on zero time, the spectrum becomes complex. This can be accomplished by changing the limits for integration from –J/2..+J/2 to –J/2 + t to J/2 + t , where the pulse occurs t1 1 1 seconds later. Thus, the spectrum of the delayed pulse is
The exponential is a helix with a pitch depending on t ,which allows the time waveform to be reconstructed uniquely1 from the spectrum. The greater t , the tighter the helix in the spectrum, as in Figure 2.27, hence the expression helical1 spectrum. The exponential term in (2.26) is in angle only, a helix of radius unity, so the modulus of the spectrum remains the same.
-0.2 0 0.5 1
-2 -1
1 2 -2
-1
1 2
-0.2 0 0.5 1
-2 -1
1 2 -2
-1
1 2
-0.5 0 0.5 1
-2 -1
1 2 -2
-1
1 2 Pulse centered on +0.5 in time
Pulse centered on zero time Pulse centered on +1 in time
Frequency
Time Time Time
Frequency Frequency
Pitch " 1/t1
Spectrum, H(f) ' %
I
4&4
h(t) exp&j 2Bft dt
Waveform, h(t) ' %
I
4
&4
H(f) exp%j 2Bft df
I
%4
&4
*(t) exp(&j 2B f t) dt ' 1
h(t) ' exp(j 2B f1 t) exp(&j 2B f t)
&4 %4
Figure 2.27 The effects of the pulse not being centered on zero.
(2.27)
(2.28)
(2.29) The convention in Figure 2.27 and the direction of the axes are common throughout this book.
In contrast to standard textbook descriptions, most practical spectra are complex. It is often useful to plot the complex spectrum of a time series in three dimensions. The phenomena and transients causing the unwanted spectral components tend to separate into the real and imaginary planes. This eases the conceptual analysis. Modern mathematical software makes this presentation possible.
The general expression for the Fourier transform used here is
and its inverse is
Notice that the –f , or frequency, convention is used [4, p. 381] so the 1/2B factor is not necessary for the inverse transform. The exponential terms represent helices. A polyphase signal is itself a helix, say, , as shown in Figure 2.28. The multiplier in the Fourier transform in (2.27) is . During its time of existence, the function H(f) will have a value only when the frequencies of the helices are the same and their sense or phase sequences are opposite.
The helical component is effectively unraveled or stopped. With a polyphase signal of frequency f Hz, this will 1 occur at +f Hz for a positive phase sequence signal and –f Hz for a negative phase sequence. The spectrum in Figure 1 1 2.29, in contrast to that in Figure 2.27, is a fine line, or delta function (zero width, undetermined height, area unity), at only the “positive” frequency. This makes delta functions difficult to draw, so that their height is conventionally their area or unity [5].
For a cosine wave of frequency f Hz, which is composed of two polyphase waveforms of opposite phase sequences, there will be two spectral components at +f Hz and –f Hz. For noise of power P W measured over a bandwidth of zero to F Hz, the spectral density over all frequency space (from to ) is P/2F W/Hz. This is discussed in Chapter 16, Transforms.
A delta function has a value only at time zero and has an area of unity. Thus,
-1 -0.5 0 0.5
1
2 4
6 8
10
-0.1
0.1 0.2
0.3
-1 -0.5 0 0.5 1
-1 -0.5
0.5 1
0.1 0.2
0.3
-1 -0.5 0 0.5 1
-1 -0.5
0.5 1
0.1 0.2
0.3
Time, t f1 = 10 Hz
Frequency, f
Real Imaginary
Time, t Real
Real
Imaginary
FOURIER TRANSFORM MULTIPLIER exp(-j 2 B f t)
TIME WAVEFORM h(t) = exp(j 2 B f1 t)
SPECTRUM Response only when the time waveform has the same frequency but opposite sequence as the Fourier transform multiplier, here 10 Hz
I
%4
&4
*(t&a) f(t) dt ' f(a)
I
%4
&4
*(t&a) exp(&j 2B f t) dt ' exp(&j 2B f t)
-1 -0.5 0 0.5 1
-10 -5
5 10 -1
-0.5 0.1
0.5 1
-1 -0.5 0 0.5 1
-10 -5
5 10 -1
-0.5
0.5 1
Dirac pulse occurring at +½ second
Dirac pulse occurring at -
½ second
Dirac pulse occurring at zero time
Time s Frequency
Hz
Pitch 2 Hz
0 0.5 1
-10 -5
5 -0.5 10
0.5
exp(±j f)
Figure 2.28 The spectrum of a 10 Hz polyphase waveform.
(2.30)
(2.31)
Figure 2.29 The helical spectra from Dirac pulses at various times.
for all frequencies as shown in the center of Figure 2.29. If the delta function occurs at time a, then [5, p. 75]
For the spectrum of a pulse occurring at a seconds, the Fourier transform is the helix
The outer portions of Figure 2.29 show the helices, , each with a pitch of 2 Hz, for displacements of +G½ second.
The transformation of a two-dimensional real (balanced even) or imaginary (balanced odd) spectrum into a helical spectrum allows the time of occurrence of all its components to be expressed uniquely. For example, if in Figure 2.29 there are two unequal components, the spectrum will be similar to a “coiled coil” in incandescent lamps.
C(>) ' I
b a
p(x) e&j2Bx> dx and p(x) ' I
%4
&4
C(>) ej2Bx> d>
0 0.2 0.4 0.6 0.8 1
1 2 3 4 0
0.1 0.2 0.3 0.4
0.6 0.8 1 1.2 1.4
Single noise pulse
Sum of 10 noise pulses Sum of 100 noise pulses
Sum of 100 noise pulses
Equivalent Gaussian distribution (dotted)
(2.32)
Figure 2.30 The normalized distributions of the sums of 1, 10, and 100 noise pulses and an equivalent Gaussian distribution.
2.8.1 Convolution and correlation
The Fourier transform, as well as the formula that links waveforms in time to their spectra, is also used in statistics.
Complex Fourier transforms may be multiplied together either simply for convolution or by the complex conjugate, as in alternating current power calculations, called correlation.
The notation in this book used for statistics is matched to that in signal theory so that the Fourier transform or characteristic function in statistics, C(>), for a probability distribution p(x) is
where the limits of integration, a and b, are over the full range of x.
The convention is different from the +i x > found in the exponent in statistical texts.
2.8.1.1 Convolution
Convolution is used in this book to find the shapes of the sums from statistical distributions and their moments, for example, statistically described signals and noise. In Figure 2.31(a) two Gaussian curves with a variance of unity are shown and the Fourier transforms are to be seen in Figure 2.31(b) with 4 cycles per frequency unit and in Figure 2.31(c) with 6 cycles per frequency unit. The curve for the Fourier transform of the sum of samples from the two curves is found by multiplication: multiply the amplitudes and add the phase angles to give a narrower transform with 10 cycles per frequency unit — see Figure 2.31(e). The final curve in Figure 2.31(g) is found by inversion and has a mean of 10 and a variance of 2, found by adding the moments of the curves in Figure 2.31(a). This is the basis for summing the means to find bias and the variances for accuracy and the shape of the resulting Figure 2.31(g) may be used to find percentiles.
The probability distributions in Chapters 11 and 12 are not symmetric, so convolution must be used to find the shapes of the distributions of the sums.
Any number of Fourier transforms may be multiplied together to give a final distribution function and moments, as with the sum of noise samples, each with an exponential distribution in Figure 2.30. The product of an infinite number of convolutions inverts to a Gaussian distribution as stated in the central limit theorem [6, p. 219], and an example is shown in Figure 2.30.
2.8.1.2 Cross-correlation
Correlation is used to match a signal with a reference copy held in memory — see Chapter 8. In this case the phases of the signals are subtracted by reversing the sign of the Q phase of the reference signal before multiplication, as in Figure 2.31(d). After multiplication the curve in Figure 2.31(f) is found with a frequency of two cycles per frequency unit or the difference of the positions of the means in Figure 2.31(a). The inverse transform, Figure 2.31(h), gives a pulse with a lag of 2 and a variance of the sum of the variances of the original curves, 2 (standard deviation /2).
2.8.1.3 Autocorrelation
When a waveform is cross-correlated with itself, the tip of the function is the classical calculation of power, and the process is called autocorrelation. Autocorrelation is also used to give the shape of an expanded pulse and its time sidelobes after pulse compression (see Chapter 8). The process can be imagined from Figure 2.31(a) if both Gaussian curves coincide so that the lag in Figure 2.31(h) is zero and the curve is symmetrical around the ordinate axis.
0.1 0.2 0.3 0.4
2 4
6 8
10
0.05 0.1 0.15
0.2 0.25
2 4
6 8
10 12
14
0.05 0.1 0.15 0.2 0.25
-2
2 4
6 8
10 12 14
(a) Two Gaussian curves centered on 4 and 6, standard deviation 1.
(e) Fourier transform of the convolution of the two curves
(c) Fourier transform of curve centered on 6
(d) Complex conjugate of Fourier transform of curve centered on 4
Multiply Multiply
(b) Fourier transform of curve centered on 4
(f) Fourier transform of the correlation of the two curves
(g) Fourier transform of the convolution of the two curves
(h) Fourier transform of the correlation of the two curves
Lag
CONVOLUTION CORRELATION
Distance between the means = Lag
Sum of means = 10
Figure 2.31 Convolution and correlation with Fourier transforms.
p(x) '
exp &x2 2F2 2BF
f(t) ' exp &4 ln(2) t J
2
' exp &2.77259 t J
2
0 0.1 0.2 0.3 0.4
-4 -3 -2 -1 1 2 3 4 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2 -1 1 2
F F
Pulse duration between 50% points JJJJ
Standard deviation
50% points at
%%%%2 ln2 F or 1.1774 F
Standard deviation 2%%%%2 ln 2 JJJJ or 0.42466 JJJJ
Gaussian distribution Gaussian pulse
ln(2)F ' 0.8325546112F 1
2 ' 0.7071067810 2
2ln(2)F ' 1.177410022F 1
2 ' 0.5 2
2ln(4)F ' 1.665109221F 1
2
' 0.7071067810 (2.33)
(2.34)
Figure 2.32 Gaussian or normal distribution and Gaussian pulse.