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# The Discrete Fourier Transform of a Windowed Sequence

DOI link for The Discrete Fourier Transform of a Windowed Sequence

The Discrete Fourier Transform of a Windowed Sequence book

# The Discrete Fourier Transform of a Windowed Sequence

DOI link for The Discrete Fourier Transform of a Windowed Sequence

The Discrete Fourier Transform of a Windowed Sequence book

## ABSTRACT

In Chapter 7 we established that after a signal x(t) is sampled, we can only hope to compute the values of X_{I}
(f) = F{x (t) P
_{Δt} (t)},which is the Fourier transform of the sampled sequence. As discussed initially in Chapter 6 and more than once in Chapter 7, whether the central period of X_{I}
(f) agrees with or closely approximates X(f)= F{x(t)} is determined by the chosen sampling rate ℝ = 1/Δt,which cannot be changed after the signal has been sampled. When they don t agree with each other, the Fourier transform of the sequence X_{I}
(f) is said to contain aliased frequencies. While we were concerned with the mathematical relationship between the sample values of X_{I}
(f) and the sample values of the signal x(t) in Chapter 7, in this chapter we are concerned with computing the numerical values of X_{I}
(f) from a nite sequence of N samples, assuming that we have some knowledge about the duration or periodicity of the signal x(t) so that we can decide on the sample size N.