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Spectral Audio Signal Processing
    Spectrum Analysis of Sinusoids
       Spectrum of a Sampled Sinusoid


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Spectrum of a Sampled Sinusoid

In the discrete-time case, we replace $ t$ by $ nT$ where $ n$ ranges over the integers and $ T$ is the sampling period in seconds:

$\displaystyle s_{\omega_0}(n) \isdef e^{j\omega_0 n T} $

It is common notational practice in signal processing to use normalized radian frequency

$\displaystyle {\tilde \omega}\isdef \omega T \;\in[-\pi,\pi). $

Thus, our sampled sinusoid becomes

$\displaystyle s_{\omega_0}(n) \isdef e^{j{\tilde \omega}_0 n}. $

It is not difficult to convert between normalized and unnormalized frequency. The use of a tilde (`$ \tilde{}$') will explicitly indicate normalization, but it may be left off as well, so that $ \omega$ may denote either normalized or unnormalized frequency.2.4

The spectrum of infinitely long discrete-time signals is given by the Discrete Time Fourier Transform (DTFT) (discussed in §2.1):

$\displaystyle S_{\omega_0}(\omega) \isdef \sum_{n=-\infty}^{\infty} s_{\omega_0}(n) e^{-j\omega n} = 2\pi\delta(\omega-\omega_0) = \delta(f-f_0) $

where now $ \delta(\omega)$ is an impulse defined for $ \omega\in[-\pi,\pi)$ or $ f\in\left[-\frac{1}{2},\frac{1}{2}\right)$, and $ \omega$ denotes normalized radian frequency. (Treatments of the DTFT invariably use normalized frequency.)

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

 

bbhattac wrote:

8/21/2008
 
Since we used a cosine function as a continuous sinusoid in the previous section, shouldn't we use a sampled version of the same signal as a "sampled sinusoid" ? Thus we should get a +ve and a -ve frequency term here as well.
 

JOS wrote:

8/21/2008
 
That's fair - I should be saying "complex sinusoid" in this section, to be more precise. Fixed in the master. - jos

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