Chapter 12: Properties of The Fourier Transform 12.A Introduction. The power of the Fourier transform derives principally from the many theorems describing the properties of the transformation operation which provide insight into the nature of physical systems. Most of these theorems have been derived within the context of communications engineering to answer questions framed like "if a time signal is manipulated in such-and-such a way, what happens to its Fourier spectrum?" As a result, a way of thinking about the transformation operation has developed in which a Fourier transform pair y(t) Y ( f ) is like the two sides of a coin, with the original time or space signal on one side and its frequency spectrum on the other. The two halves of a Fourier transform pair are thus complementary views of the same signal and so it makes sense that if some operation is performed on one half of the pair, then some equivalent operation is necessarily performed on the other half. Many of the concepts underlying the theorems and properties described below were introduced in Chapter 6 in the context of Fourier series. For the most part, these theorems can be extended into the domain of the Fourier transform simply by examining the limit as the length of the observation interval for the signal grows without bound. Consequently, it will be sufficient here simply to list the results. Rigorous proofs of these theorems may be found in most standard textbooks (e.g. Bracewell). 12.B Theorems Linearity Scaling a function scales it's transform pair. Adding two functions corresponds to adding the two frequency spectra. If h( x ) H ( f ) then ah( x ) aH ( f ) [12.1] h( x ) H ( f ) If then h( x ) + g( x ) H ( f ) + G( f ) [12.2] g( x ) G ( f ) Scaling Multiplication of the scale of the time/space reference frame changes by the factor s inversely scales the frequency axis of the spectrum of the function. If h( x ) H ( f ) then h( x / s ) s H( f s) [12.3] and s h( x s) H( f / s ) [12.4] Chapter 12: Properties of The Fourier Transform Page 109 A simple, but useful, implication of this theorem is that if h( x ) H ( f ) then h( -x ) H( - f ) . In words, flipping the time function about the origin corresponds to flipping its spectrum about the origin. Notice that this theorem differs from the corresponding theorem for discrete spectra (Fig. 6.3) in that the ordinate scales inversely with the abscissa. This is because the Fourier transform produces a spectral density function rather than a spectral amplitude function, and therefore is sensitive to the scale of the frequency axis. Time/Space Shift Displacement in time or space induces a phase shift proportional to frequency and to the amount of displacement. This occurs because a given displacement represents more cycles of phase shift for a high-frequency signal than for a low- frequency signal. h( x ) H ( f ) h( x - x 0 ) e- i2 fx 0 If then H( f ) [12.5] Frequency Shift Displacement in frequency multiplies the time/space function by a unit phasor which has angle proportional to time/space and to the amount of displacement. h( x ) H ( f ) h( x )ei2 H( f - f0 ) xf0 If then [12.6] Modulation Multiplication of a time/space function by a cosine wave splits the frequency spectrum of the function. Half of the spectrum shifts left and half shifts right. This is simply a variant of the shift theorem which makes use of Euler's relationship cos( x ) = (e ix + e -ix )/ 2 h( x )ei2 H( f - f0 ) xf0 if h( x ) H ( f ) then h( x)e -i 2 H( f + f0 ) xf0 and therefore by the linearity theorem it follows that H ( f - f0 ) + H ( f + f 0 ) h( x )cos(2 xf0 ) 2 H ( f - f 0) + H ( f + f0 ) h( x )sin(2 xf0 ) [12.7] 2i The modulation theorem is the basis of transmission of amplitude-modulated radio broadcasts. When a low frequency audio signal is multiplied by a radio- frequency carrier wave, the spectrum of the audio message is shifted to the radio portion of the electromagnetic spectrum for transmission by an antenna. Chapter 12: Properties of The Fourier Transform Page 110 Similarly, a method for recovery of the audio signal called super-heterodyne demodulation involves multiplying the received signal by a sinusoid with the same frequency as the carrier, thereby demodulating the audio component. Differentiation Differentiation of a function induces a 90 phase shift in the spectrum and scales the magnitude of the spectrum in proportion to frequency. Repeated differentiation leads to the general result: d n h( x) If h( x ) H ( f ) then n (i 2 f )n H ( f ) [12.8] dx This theorem explains why differentiation of a signal has the reputation for being a noisy operation. Even if the signal is band-limited, noise will introduce high frequency signals which are greatly amplified by differentiation. Integration Integration of a function induces a -90 phase shift in the spectrum and scales the magnitude of the spectrum inversely with frequency. x If h( x ) H ( f ) then - h(u ) du H ( f )/(i 2 f ) + constant [12.9] From this theorem we see that integration is analagous to a low-pass filter which blurs the signal. Transform of a transform We normally think of using the inverse Fourier transform to move from the frequency spectrum back to the time/space function. However, if instead the spectrum is subjected to the forward Fourier transform, the result is a time/space function which has been flipped about the y-axis. This gives some appreciation for why the kernels of the two transforms are complex conjugates of each other: the change in sign in the reverse transform flips the function about the y-axis a second time so that the result matches the original function. h(t ) H( f ) H( f ) h(-t ) F F If then [12.10] One practical implication of this theorem is a 2-for-1 bonus: every transform pair brings with it a second transform pair at no extra cost. If h(t ) H ( f ) then H(t ) h( - f ) [12.11] For example, rect( t ) sinc( f ) implies sinc(t) rect(- f ) . This theorem highlights the fact that the Fourier transform operation is fundamentally a mathematical relation that can be completely divorced from the physical notions of time and frequency. It is simply a method for transforming a Chapter 12: Properties of The Fourier Transform Page 111 function of one variable into a function of another variable. So, for example, in probability theory the Fourier transform is used to convert a probability density function into a moment-generating function, neither of which bear the slightest resemblance to the time or frequency domains. Central ordinate By analogy with the mean Fourier coefficient a0 , the central ordinate value H(0) (analog of a0 in discrete spectra) represents the total area under the function h(x). - h(u) e du = -i 0 If h( x ) H ( f ) then H(0) = h(u) du - h(0) = - H (u) ei 0du For the inverse transform, = - H(u) du [12.12] = - Re[H (u )] du + i - Im[ H(u )] du Note that for a real-valued function h(t) the imaginary portion of the spectru will have odd symmetry, so the area under the real part of the spectrum is all that needs to be computed to find h(0). For example, in optics the line-spread function (LSF) and the optical transfer function (OTF) are Fourier transform pairs. Therefore, according to the central- ordinat theorem, the central point of the LSF is equal to the area under the OTF. In two dimensions, the transform relationship exists between the point-spread function (PSF) and the OTF. In such 2D cases, the integral must be taken over an area, in which case the result is interpreted as the volume under the 2D surface. Equivalent width A corollary the central ordinate theorm is h(u ) du = H(0) If - then - h(u ) du = H (0) [12.13] - H (u) du h(0) = H (u) du h(0) - The ratio on the left side of this last expression is called the "equivalent width" of the given function h because it represents the width of a rectangle with the same central ordinate and the same area as h. Likewise, the ratio on the right is the inverse of the equivalent width of H. Thus we conclude that the equivalent width of a function in one domain is the inverse of the equivalent width in the other domain as illustrated in Fig. 12.1. For example, as a pulse in the time domain gets shorter, its frequency spectrum gets longer. This theorem quantifies that relationship for one particular measure of width. Chapter 12: Properties of The Fourier Transform Page 112 Fig. 12.1 Equivalent Width Theorem Space/time Domain Frequency Domain w 1 w Convolution The convolution operation (denoted by an asterisk) is a way of combining two functions to produce a new function. By definition, p = hg means p( x ) = - g(u)h( x - u ) du [12.14] Convolution will be described in detail in section 12C. Here it is sufficient to state the convolution theorem: h( x ) H ( f ) h( x ) g( x) H ( f ) G( f ) If then [12.15] g( x ) G ( f ) h( x ) g( x) H( f ) G( f ) In words, this theorem says that if two functions are multiplied in one domain, then their Fourier transforms are convolved in the other domain. Unlike the cross-correlation operation described next, convolution obeys the commutiative, associative, and distributive laws of algebra. That is, commutative law hg = gh associative law f (g h) = ( f g) h [12.16] distributive law f ( g + h) = f g + f h Derivative of a convolution Combining the derivative theorem with the convolution theorm leads to the conclusion dh df dg If h( x ) = f ( x) g(x ) then = g = f [12.17] dx dx dx In words, this theorem states that the derivative of a convolution is equal to the convolution of either of the functions with the derivative of the other. Cross-correlation The cross-correlation operation (denoted by a pentagram) is a way of combining two functions to produce a new function that is similar to convolution. By definition,Download Link: