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Extra resources for DSP Lectures

Example text

The central x(t)1 pulse on the t = 0 axis remains, but all of its replicas are removed to infinity, and then X(f) becomes this pulse's spectrum: 60 New Page 1 Meanwhile, the harmonics crowd ever more closely together: 1 As k⋅ε → f, this summation becomes an integral, and all that remains of x(t)~ is the central x(t) pulse: Combining our two results: Forward CFT Inverse CFT This is a new transform, a Continuous Fourier Transform (CFT), and it links a pulse−shaped time signal x(t)1 to a pulse−shaped spectrum X(f).

We will use it to derive the CFT (continuous in both domains), that links a pulse in time to a pulse−shaped spectrum. 1 From DfFT to CFT The DfFT ( using ε = 1/P ) took the form: Forward DfFT Inverse DfFT 57 New Page 1 We've included ~ in x(t)~ to mark its periodicity. In this chapter, the simpler x(t) will mean a pulse−shaped signal. The forward DfFT integral above refers to the harmonic frequencies f = kε, and we can think of C k as being samples from a spectral function X(f) taken at the frequencies f = kε, where: and then: with ε = 1/P These Ck are area−samples of X(f) (value−samples X(kε) × sample−spacing ε).

They are the area−samples {ε⋅X(kε)} of X(f). In this way, the DfFT is just a special (impulsive) version of the CFT. 2 CFT Pairs: rectangle and sinc A pulse in time has a pulse−shaped CFT spectrum. As an example of this, we will find the CFT spectrum of the rectangle pulse x(t) = Π(t) shown here (Fig ç ). The integration is easy: 63 New Page 1 The integral of eat is eat/a (whether a is real or complex), and so: The result is sin(πf)/(πf), a well−known shape (Fig ê ), which we call a "sinc" pulse.