5.2 Discrete Transformations of Signal Processing

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the discrete periodic signal can be described like an analogue signal as a sum of e-

functions according to

fp(i) =

N−1

∑

l=0

cle^j2πil/N ,

i = 0, 1, . . . , N −1

(5.19)

describe. The Fourier-coefficients ci are then given by

cl = ¹

N

N−1

∑

i=0

fp(i)e^−j2πil/N ,

l = 0, 1, . . . , N −1 .

(5.20)

A comparison with Equation 5.18 shows that the coefficients of the discrete Fourier-

transform match the values of the discrete-time Fourier-transform FD(l/NTa) except

for the factor 1/N:

cl = ¹

N ^{FD (}

l

NTa

)

bzw.

fp(i) = ¹

N

N−1

∑

l=0

FD (

l

NTa

) e^j2πil/N .

(5.21)

Result

By periodically continuing a time-limited signal sampled with the time interval Ta

with a period NTa greater than the temporal length tg < NTa of the signal, the associ-

ated spectrum can be calculated by a discrete Fourier series according to Equation 5.16

and Equation 5.19 at N frequency points within one period of the frequency domain.

The original signal can be realised by cutting out one period of the signal fp(i) by

a multiplication with a temporal rectangular window, and the original spectrum can

be interpolated by interpolation in the frequency domain with an si function, which

corresponds to a convolution of the discrete spectrum F(l/NTa) with the Fourier-

transform of the temporal window. With the help of the abbreviation

F(l) := FD (

l

NTa

) =

N−1

∑

i=0

fp(i)e^−j2πil/N

(5.22)

and Equation 5.19 are finally obtained for the outward and backward transformation

of the discrete Fourier-transformation (DFT):

fp(i) = ¹

N

N−1

∑

l=0

F(l)e^j2πil/N ,

F(l) =

N−1

∑

i=0

fp(i)e^−j2πfil/N .

(5.23)

The DFT can also be given in matrix notation using the Fourier-matrix W := {wmn},

with elements wmn := e^−j2πmn/N as follows: ³

f p = W⁻¹ ⋅F

F = W ⋅fp

(5.24)

3 The matrix A^󸀠is the transpose of the matrix A. Given a vector, e.g. fp, this gives a row vector a column

vector and a column a row vector.