Discrete-time Fourier Transform - Relationship To Sampling

Relationship To Sampling

Often the x sequence represents the values (aka samples) of a continuous-time function, x(t), at discrete moments in time: t = nT, where T is the sampling interval (in seconds), and is the sampling rate (samples per second). Then the DTFT provides an approximation of the continuous-time Fourier transform:

To understand this, consider the Poisson summation formula, which indicates that a periodic summation of function X(f) can be constructed from the samples of function x(t). The result is:

X_{1/T}(f)\ \stackrel{\mathrm{def}}{=} \sum_{k=-\infty}^{\infty} X\left(f - k/T\right) \equiv \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x}\ \underbrace{e^{-i 2\pi f T n}}_{\mathcal{F}\{\delta(t-nT)\}}\ =\ \mathcal{F}\left \{\sum_{n=-\infty}^{\infty} x \cdot \delta(t-nT)\right \}

(Eq.2)

comprises exact copies of that are shifted by multiples of ƒs and combined by addition. For sufficiently large ƒs, the k=0 term can be observed in the region with little or no distortion (aliasing) from the other terms.

With the following definition of a normalized frequency, the expressions in Eq.2 and Eq.1 are identical:

Since represents ordinary frequency (cycles per second) and has units of samples per second, the units of are cycles per sample. It is common notational practice to replace this ratio with a single variable, which represents actual frequencies as multiples (usually fractional) of the sample rate. In terms of that normalization, the periodicty of the DTFT is 1. ω is also a normalized frequency, with units of radians per sample. And in terms of ω, the periodicity is 2π.

Read more about this topic:  Discrete-time Fourier Transform

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