Spectrum of a Sinusoid
A sinusoid is any signal of the form
where is the amplitude (in arbitrary units), is the phase in radians, and is the frequency in radians per second. Time is a real number that varies continuously from minus infinity to infinity in the ideal sinusoid. All three parameters are real numbers. In addition to radian frequency , it is useful to define , where is the frequency in Hertz (Hz).6.1
By Euler's identity, , we can write
where denotes the complex conjugate of . Thus, we can build a real sinusoid as a linear combination of positive- and negative-frequency complex sinusoidal components:
The spectrum of is given by its Fourier transform (see §2.2):
In this case, is given by (5.2) and we have
We see that, since the Fourier transform is a linear operator, we need only work with the unit-amplitude, zero-phase, positive-frequency sinusoid . For , may be called the analytic signal corresponding to .6.2
It remains to find the Fourier transform of :
where is the delta function or impulse at frequency (see Fig.5.4 for a plot, and §B.10 for a mathematical introduction). Since the delta function is even ( ), we can also write . It is shown in §B.13 that the sinc limit above approaches a delta function . However, we will only use the Discrete Fourier Transform (DFT) in any practical applications, and in that case, the result is easy to show .
The inverse Fourier transform is easy to evaluate by the sifting property6.3of delta functions:
Substituting into (5.4), the spectrum of our original sinusoid is given by
which is a pair of impulses, one at frequency having complex amplitude , summed with another at frequency with complex amplitude .
Spectrum of Sampled Complex Sinusoid
Optimal FIR Digital Filter Design