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=0,elsewhere between-T/2 and T/2 (6-1) The Fourier Series for y(t) is defined as yt =c k exp ik2πt T k=-∞ (6-2) with c k 1 T yt exp -ik2πt T dt -T/2 T/2 (6-3) Evaluating this integral gives c k 1 T Aexp -ik2πt T dt -τ/2

τ/2

A T T -ik2π exp -ik2πt T -τ/2

τ/2

A T T kπ sin kπτ T Aτ T sin kπτ T kπτ T Aτ T sinc kπτ T

(6-4) where we have used the relationship exp(iy) = cos(y) + i sin(y) to evaluate the integral with the cosine terms cancelling because of symmetry. The Fourier Series coefficients for a pulse train is given by a sinc function.

ESS 522 2014 6-2 The largest amplitude terms in the Fourier series have k < T/τ. Also if T = τ then the time series has a constant amplitude and all the coefficients except c0 are equal to zero (the equivalent of the inverse Fourier transform of a Dirac delta function in frequency). Now if we allow each pulse to become a delta function which can be written mathematically by letting τ → 0 with A = 1/τ which yields a simple result c

k 1 T ,limτ→0,A=1/τ

(6-5) A row of delta functions in the time domain spaced apart by time T is represented by a row of delta functions in the frequency domain scaled by 1/T and spaced apart in frequency by 1/T (remember f = k/T). Our row of equally spaced pulses is known as a Dirac comb. If we a define a Dirac comb in the time domain with the notation C(t,T) such that C(t,T)=δt-kT

k=-∞ , (6-6) then its Fourier transform is another Dirac comb function. FTC(t,T) 1 T Cf, 1 T =g 0 (t)*C(t,T) (6-8) Convolution in the time domain is multiplication in the frequency domain so we can write Gf =G 0 f .FTCt;T =G 0 f 1 T Cf; 1 T

(6-9) The spectrum of the periodic function (Figure 1) is just a sampled version of the continuous spectrum of g0(t) with the samples scaled by a constant 1/T. A periodic continuous function time has a discrete frequency spectrum. Sampling Operator. If we take our function g0(t) and multiply it by a Dirac comb C(t,Δt) we obtain a sampled version g0(t) which we denote by gs(t) (Figure 2). g

s t =g 0 t .Ct;Δt (6-10) Multiplication in the time domain is convolution in the frequency domain so we can write G s f =G 0 f *FTCt,Δt 1 Δt G 0 f *Cf, 1 Δt (6-11)

ESS 522 2014 6-3 The frequency spectrum of Gs(f) is scaled by 1/Δt and replicated at intervals of 1/Δt (Figure 1). A discrete continuous function of time has a periodic frequency spectrum. As we discussed in the lecture 5, we see again that sampling a time series removes our ability to discriminate between frequencies spaced at intervals of 1/Δt. If we know our time series is limited to a maximum frequency f

max 1

2Δt

=f

Nyquist

(6-12) then there is no ambiguity in the frequency domain since the replicated spectra do not overlap. However is fmax > fNyquist then the replicated spectra do overlap additively and we cannot discriminate between them in the frequency domain. Recovery of a continuous time signal from a sampled time series We have seen that if we sample at a frequency that is not at least twice the maximum frequency of the signal then we loose information. An interesting corollary of this is that if we sample a continuous signal that is band limited below the Nyqyist frequency then we can unambiguously recover the continuous signal from the sampled signal. This is known as Shannon's sampling theorem. To show this, consider a repetitive spectrum that is obtained from a sampled time series and multiply it by a boxcar function of height Δt and width 1/ Δt. Provided the frequency content of the original signal is band-limited below the Nyquist frequency this gives us G0(f). G

0 f =G S f .B c f, -1

2Δt

1

2Δt

Δt

(6-13) where Bc (f,a,b) is a boxcar which has value of 1 for a < f < b and 0 elsewhere. In the time domain we can write g

0 t =FT -1 G 0 f =g 0 t

δt-kΔt

k=-∞ *Δtexpi2πft df 1

2Δt

1

2Δt

(6-14) Now we can evaluate the integral Δtexpi2πft df 1

2Δt

1

2Δt

Δt i2πf expi2πft 1

2Δt

1

2Δt

Δt i2πf

2sinsin

πt Δt sin πt Δt πt Δt =sinc πt Δt (6-15) Equation (6-14) becomes g 0 t =FT -1 G 0 f =g 0 t

δt-kΔt

k=-∞ *sinc πt Δt (6-16) Convolution is defined by the following

ESS 522 2014 6-4 at

*bt =aτ bt-τ dτ (6-17) So we can write g 0quotesdbs_dbs31.pdfusesText_37

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