0.3 Sampling, up--sampling, down--sampling, and multi--rate (Page 4/5)

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Calculation of the fourier transform and fourier series using the fft

Most theoretical and mathematical analysis of signals and systems use the Fourier series, Fourier transform, Laplace transform, discrete-timeFourier transform (DTFT), or the z-transform, however, when we want to actually evaluate transforms, we calculate values at sample frequencies.In other words, we use the discrete Fourier transform (DFT) and, for efficiency, usually evaluate it with the FFT algorithm. An importantquestion is how can we calculate or approximately calculate these symbolic formula-based transforms with our practical finite numerical tool. Itwould certainly seem that if we wanted the Fourier transform of a signal or function, we could sample the function, take its DFT with the FFT, andhave some approximation to samples of the desired Fourier transform. We saw in the previous section that it is, in fact, possible provided somecare is taken.

Summary

For the signal that is a function of a continuous variable we have

\begin{matrix} FT: & f (t) & \to & F (ω) \\ DTFT: & f (T n) & \to & \frac{1}{T} F_{p} (ω) = \frac{1}{T} \sum_{ℓ} F (ω + 2 π ℓ / T) \\ DFT: & f_{p} (T n) & \to & \frac{1}{T} F_{p} (Δ k) for Δ T N = 2 π \end{matrix}

For the signal that is a function of a discrete variable we have

\begin{matrix} DTFT: & h (n) & \to & H (ω) \\ DFT: & h_{p} (n) & \to & H (Δ k) for Δ N = 2 π \end{matrix}

For the periodic signal of a continuous variable we have

\begin{matrix} FS: & \tilde{g} (t) & \to & C (k) \\ DFT: & \tilde{g} (T n) & \to & N C_{p} (k) for T N = P \end{matrix}

For the sampled bandlimited signal we have

\begin{matrix} Sinc: & f (t) & \to & f (T n) \\ f (t) = \sum_{n} f (T n) sinc (2 π t / T - π n) \\ if F (ω) = 0 for | ω | > 2 π / T \end{matrix}

These formulas summarize much of the relations of the Fourier transforms of sampled signals and how they might be approximately calculate with theFFT. We next turn to the use of distributions and strings of delta functions as tool to study sampling.

Sampling functions — the shah function

Th preceding discussions used traditional Fourier techniques to develop sampling tools. If distributions or delta functions are allowed, theFourier transform will exist for a much larger class of signals. One should take care when using distributions as if they were functions but itis a very powerful extension.

There are several functions which have equally spaced sequences of impulses that can be used as tools in deriving a sampling formula. Theseare called “pitch fork" functions, picket fence functions, comb functions and shah functions. We start first with a finite length sequence to beused with the DFT. We define

⨿_{M} (n) = \sum_{m = 0}^{L - 1} δ (n - M m)

where $N = L M$ .

D F T {⨿_{M} (n)} = \sum_{n = 0}^{N - 1} [\sum_{m = 0}^{L - 1}, δ, (n - M m)] e^{- j 2 π n k / N}

= \sum_{m = 0}^{L - 1} [\sum_{n = 0}^{N - 1}, δ, (n - M m), e^{- j 2 π n k / N}]

= \sum_{m = 0}^{L - 1} e^{- j 2 π M m k / N} = \sum_{m = 0}^{L - 1} e^{- j 2 π m k / L}

= {\begin{matrix} L & < k > L = 0 \\ 0 & otherwise \end{matrix}

= L \sum_{l = 0}^{M - 1} δ (k - L l) = L ⨿_{L} (k)

For the DTFT we have a similar derivation:

D T F T {⨿_{M} (n)} = \sum_{n = - \infty}^{\infty} [\sum_{m = 0}^{L - 1}, δ, (n - M m)] e^{- j ω n}

= \sum_{m = 0}^{L - 1} [\sum_{n = - \infty}^{\infty}, δ, (n - M m), e^{- j ω n}]

= \sum_{m = 0}^{L - 1} e^{- j ω M m}

= {\begin{matrix} L & ω = k 2 π / M \\ 0 & otherwise \end{matrix}

= \sum_{l = 0}^{M - 1} δ (ω - 2 π l / M l) = K ⨿_{2π/M} (ω)

where $K$ is constant.

An alternate derivation for the DTFT uses the inverse DTFT.

I D T F T {⨿_{2 π / M} (ω)} = \frac{1}{2 π} \int_{- π}^{π} ⨿_{2 π / M} (ω) e^{j ω n} d ω

= \frac{1}{2 π} \int_{- π}^{π} \sum_{l} δ (ω - 2 π l / M) e^{j ω n} d ω

= \frac{1}{2 π} \sum_{l} \int_{- π}^{π} δ (ω - 2 π l / M) e^{j ω n} d ω

= \frac{1}{2 π} \sum_{l = 0}^{M - 1} e^{2 π l n / M} = {\begin{matrix} M / 2 π & n = M \\ 0 & otherwise \end{matrix}

= (\frac{M}{2π}) ⨿_{2π/M} (ω)

Therefore,

⨿_{M} (n) \to (\frac{2π}{M}) ⨿_{2π/T} (ω)

For regular Fourier transform, we have a string of impulse functions in both the time and frequency. This we see from:

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Source: OpenStax, Brief notes on signals and systems. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10565/1.7

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