Consider the continuous function f(t) = sin(2πnt).(a) What is the period of f(t)?(b) What...

Consider the continuous function f(t) = sin(2πnt).

(a) What is the period of f(t)?

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(b) What is the frequency of f(t)?

The Fourier transform, F(µ), of f(t) is purely imaginary (Problem 4.3), and because the transform of the sampled data consists of periodic copies of F (µ), the transform of the sampled data, , will also be purely imaginary. Draw a diagram similar to Fig. 4.6, and answer the following questions based on your diagram (assume that sampling starts at t = 0).

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(c) What would the sampled function and its Fourier transform look like in general if f(t) is sampled at a rate higher than the Nyquist rate?

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(d) What would the sampled function look like in general if f(t) is sampled at a rate lower than the Nyquist rate?

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(e) What would the sampled function look like if f(t) is sampled at the Nyquist rate with samples taken at t = 0, ΔT, 2 ΔT,... ?

FIGURE 4.6

(a) Fourier transform of a band-limited function.

(b)–(d) Transforms of the corresponding sampled function under the conditions of over-sampling, criticallysampling, and under-sampling, respectively.

4.3 It can be shown (Bracewell [2000]) that 1 ⇔ δ(µ) and δ(t) ⇔ 1. Use the first of these properties and the translation property from Table 4.3 to show that the Fourier transform of the continuous function f(t) = sin(2πnt), where n is a real number,is F(µ) = (j/2)[δ(µ + n) − δ(µ − n)].

TABLE 4.3

Summary of DFT pairs.The closedform expressions in 12 and 13 are valid only for continuous variables.They can be used with discrete variables by sampling the closed-form, continuous expressions.