Spectrum of Sampled Complex Sinusoid

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Spectral Audio Signal Processing

Spectrum Analysis of Sinusoids

Spectrum of a Sinusoid

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The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

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A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

The sampling interval (or sampling period) of a discrete-time signal is defined as the reciprocal of the sampling rate. Since the sampling rate is normally in units of samples per second (Hz), the sampling interval is normally in units of seconds. See also: sampling rate, sampling theorem — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Spectrum Analysis of Sinusoids

Spectrum of a Sinusoid

Spectrum of a Windowed Sinusoid

Spectrum of Sampled Complex Sinusoid

In the discrete-time case, we replace by where ranges over the integers and is the sampling period in seconds. Thus, for the positive-frequency component of the sinusoid of the previous section, we obtain

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

(6.8)

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

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

(6.9)

Thus, our sampled complex sinusoid becomes

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

(6.10)

It is not difficult to convert between normalized and unnormalized frequency. The use of a tilde (` $\tilde{\null}$ ') will explicitly indicate normalization, but it may be left off as well, so that $\omega$ may denote either normalized or unnormalized frequency.^6.4

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

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

(6.11)

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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Spectrum of Sampled Complex Sinusoid

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University