08-08-2012, 04:38 PM
Discrete Fourier transform
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Continuous to discrete Fourier transform
In order to motivate the discussion of the DFT, let us assume that we are
interested in computing the CTFT of a CT signal x(t) using a digital computer.
The three main steps involved in the computation of the CTFT are illustrated in
Fig. 12.1. The waveforms for the CT signal x(t) and its CTFT X(), shown in
Figs 12.1(a) and (b), are arbitrarily chosen, so the following procedure applies
to any CT signal. A brief explanation of each of the three steps is provided
below.
Discrete Fourier transform
aforementioned distortions would always be present when approximating the
CTFT with the DFT. This implies that Eq. (12.12) is an approximation for the
CTFT X() that, even at its best, only leads to a near-optimal estimation of the
spectral content of the CT signal.
On the other hand, the DFT representation provides an accurate estimate of
the DTFT of a time-limited sequence x[k] of length N. By comparing the DFT
spectrum, Fig. 12.1(h), with the DFT spectrum, Fig. 12.1®, the relationship
between the DTFT X2() and the DFT X2[r] is derived. Except for a factor of
K/M, we note that X2[r ] provides samples of the DTFT at discrete frequencies
r = 2r/M, for 0 ≤ r ≤ (M−1). The relationship between the DTFT and
DFT is therefore given by
Spectrum analysis using the DFT
In this section, we illustrate how the DFT can be used to estimate the spectral
content of the CT and DT signals. Examples 12.6–12.8 deal with the CT signals,
while Examples 12.9 and 12.10 deal with the DT sequences.
Zero padding
To improve the resolution of the frequency axis in the DFT domain, a commonly
used approach is to append theDTsequences with additional zero-valued
samples. This process is called zero padding, and for an aperiodic sequence x[k]
of length N is defined as follows: