A repertoire of DSP transforms, especially sine & cosine transforms & their significance —- part two

Just onc comment before we delve further: the trick in DSP and Digital Control and Digital Communications is to treat all possible practical signals via a few elementary or test signals. The \delta function is an elementary signals, so also are the complex exponentials or sinusoids, the unit step function, the boxcar, the  sinc pulse, and the signum function.

We continue our light study with transforms of sinusoids.

The Fourier transform of sines and cosines plays a central role in the study of linear time invariant systems for three quite related reasons: (1) they are eigenfunctions of LTI systems (2) the spectrum consists of the eigenvalues associated with these eigenfunctions, and (3) the weighting factor (sometimes called the kernel) in the Fourier transform is a complex sinusoid. Clearly, we want to include sinusoids in our Fourier transform repertoire.

We have already seen that the Fourier transform of the complex exponential is a \delta function located at the center of the complex sinusoid. That is,

\int_{-\infty}^{\infty}e^{i \omega_{0} t}e^{-i \omega t}dt=2 \pi \delta (\omega - \omega_{0})

and letting \omega_{0} \rightarrow -\omega_{0}

we get \int_{-\infty}^{\infty}e^{-i \omega_{0} t}e^{i \omega t}dt which equals

2\pi \omega (\omega + \omega_{0}).

The Fourier transform of the cosine and the sine follows directly from these two equations. Adding the two equations gives

\int_{-\infty}^{\infty}\cos (\omega_{0} t)e^{-i \omega t}dt=\pi [\delta (\omega + \omega_{0})+\delta (\omega - \omega_{0})] Equation I 

while subtracting them gives

\int_{-\infty}^{\infty}\sin (\omega_{0}t)e^{-i \omega t}dt=\pi [\delta (\omega +\omega_{0})-\delta (\omega - \omega_{0}] Equation II

These transforms are just the combinations of \delta functions that are required by the basic symmetry properties of the Fourier transform; because the cosine is a real and even function, its transform is real and symmetric. The sinc is real and odd; therefore, it has an odd imaginary transform.

Alternately, we could derive the sine and cosine transforms by using the phase-shift theorem; shifting a function along the axis in one domain introduces a complex sinusoid in the other domain. For example, if we want to generate the dual pairs in equations I and II, we apply the phase shift theorem to the \delta function and write

FT(\delta (t-t_{0}))=e^{-i\omega t_{0}}

and FT (\delta (t+t_{0}))=e^{i \omega_{0} \omega}

Adding and subtracting these two equations gives

FT(\delta (t-t_{0}) ) + \pi \delta(t + t_{0}) = 2 \cos (\omega t_{0}) equation III

FT(\delta (t+t_{0}) + \pi \delta (t-t_{0}) = i2 \sin (\omega t_{0}). equation IV

The sine and the cosine transforms and their duals are shown in the attached figure 1.

Thus, the Fourier transforms of sines and cosines can be viewed as resulting from forcing certain symmetries into  the \delta function transform after it is shifted along the axis; shifting the \delta function off the origin in the frequency domain and then requiring a symmetric symmetric spectrum results in equation I. An antisymmetric spectrum leads to Equation II. Analogous statements apply to equations for \delta function shifts along the time axis.

The \delta functions in Equations I and II make for easy convolutions leading to an important observation. For example, convolving equation I in the frequency domain with a given function F(\omega) and approximately multiplying by f(t) in the time domain gives

FT(f(t)\cos (\omega_{0} t)) = \pi (F(\omega + \omega_{0})+F(\omega - \omega_{0})) equation V

This equation and its sine wave counterpart are sometimes referred to as the modulation theorem. They show that modulating the amplitude of a function by a sinusoid shifts the function’s spectrum to higher frequencies by an amount equal to the modulating frequency. . This effect is well understood in television and radio work where a sinusoidal carrier is modulated by the program signal. Obtaining transforms of sines, cosines, and related functions has been a good example of exploiting the basic Fourier transform properties. Our next example is no so obliging.

More later…

Nalin Pithwa

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