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Chapter 12: Problem 4

Prove the similarity relation that if \(F(\omega)\) is the Fourier transform of \(f(t)\) then \((1 / a) F(\omega / a)\) is the Fourier transform of \(f(a t)\), where \(a\) is a real, positive constant.

Short Answer

Expert verified
Question: Prove that if \(F(\omega)\) is the Fourier transform of \(f(t)\), then \((1/a)F(\omega/a)\) is the Fourier transform of \(f(at)\), where \(a\) is a real, positive constant. Answer: To prove this, we first applied the Fourier transform definition to \(f(at)\), obtaining an integral expression for its Fourier transform \(G(\omega)\). Then, we used the substitution \(u = at\) to simplify the integral. Finally, we observed that the resulting integral matches the Fourier transform definition, allowing us to write \(G(\omega) = (1/a)F(\omega/a)\), which proves the similarity relation for Fourier transforms.

Step by step solution

01

Definition of the Fourier Transform and its inverse

The Fourier transform of a function \(f(t)\) is given by: $$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j \omega t} dt $$ And the inverse Fourier transform is given by: $$ f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} d\omega $$ In this exercise, we want to prove that the Fourier transform of \(f(at)\) is equal to \((1/a)F(\omega/a)\).
02

Apply the Fourier Transform to \(f(at)\)

Let \(G(\omega)\) denote the Fourier transform of \(f(at)\). Using the Fourier transform definition, we obtain: $$ G(\omega) = \int_{-\infty}^{\infty} f(at) e^{-j\omega t} dt $$
03

Apply the substitution \(u = at\)

We will now make the substitution \(u = at\), \(t = u/a\), and \(dt=du/a\). Plugging this into the integral, we get: $$ G(\omega) = \int_{-\infty}^{\infty} f(u) e^{-j\omega\frac{u}{a}}\frac{du}{a} $$ By factoring the \(1/a\) from the integral, we obtain: $$ G(\omega) = \frac{1}{a} \int_{-\infty}^{\infty} f(u) e^{-j\frac{\omega}{a}u} du $$
04

Recognize the Fourier Transform

Notice that the integral in the expression for \(G(\omega)\) is the Fourier transform of \(f(u)\). Therefore, we can express \(G(\omega)\) as follows: $$ G(\omega) = \frac{1}{a}F\left(\frac{\omega}{a} \right) $$
05

Conclusion

We have shown that if \(F(\omega)\) is the Fourier transform of \(f(t)\), then \((1/a)F(\omega/a)\) is the Fourier transform of \(f(at)\), where \(a\) is a real, positive constant. This proves the similarity relation for Fourier transforms.

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Most popular questions from this chapter

Find, using (12.2.2), the Fourier transform \(F(\omega)\) of a single rectangular pulse of amplitude \(a\) extending from \(-\tau\) to \(+\tau\). Substitute this \(F(\omega)\) in (12.2.1) and compare the resulting expression with (12.1.8), obtained from the Fourier series expansion of a sequence of rectangular pulses.

Show that the Fraunhofer diffraction integral for a one-dimensional aperture, such as a single slit (Sec. 10.4) or a grating (Secs. \(10.7\) and 10.8) may be expressed in the form $$ \psi(s)=\breve{C} \int_{-\infty}^{\infty} \breve{T}(x) e^{i \theta x} d x $$ where \(s \equiv \kappa\left(\sin \theta_{o}+\sin \theta_{o}\right)\) and \(\breve{T}(x)\) is the aperture transmission function (9.4.19), possibly \(\dagger\) For further details concerning holography and other applications of Fourier analysis to optical problems, see M. Born and E. Wolf, "Principles of Optics," 3 ded., sec. \(8.10\), Pergamon Press, New York, 1965; A. Papoulis, "Systems and Transforms with Applications in Optics," McGraw-Hill Book Company, New York, 1968; and J. W. Goodman, "Introduction to Fourier Optics," McGraw-Hill Book Company, New York, \(1968 .\) complex, that expresses the wave- transmission properties of the aperture. Hence show that the diffracted-wave amplitude, except for a constant factor, is the Fourier transform of the aperture transmission function. It is thus no accident that the Fourier transform of a rectangular pulse (Prob. \(12.22\) ) is functionally the same as the Fraunhofer diffraction-amplitude function for a single slit, \(\sin u / u\).

A pulse described by the Dirac delta function \(\delta(t)\) has negligible duration, infinite height at \(t=0\), but unit area. Find the amplitude spectrum of \(\delta(t)\) as a limiting case by setting the area of the rectangular pulse in Prob. 1222 equal to unity and then letting \(\tau \rightarrow 0 .\) Answer: \(F(\omega)=1\).

A sinusoidal voltage \(V=V_{0} \cos \omega t\), whose frequency can be varied, is applied to the terminals of a network, and the component of the current in phase with \(V\) is found to be \(I_{r}=\left[V_{0} / R\left(1+\omega^{2} \tau^{2}\right)\right] \cos \omega t .\) Use (12.7.15) to find the out-of-phase component of current. What is the input impedance of the network?

Discuss the need for an amplifier to have a certain minimum bandwidth \(\Delta \omega\) if it is to be used for amplifying pulses of duration \(\Delta t\).
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