The Fourier transform of a continuous signal

The approximation is still imperfect and can be improved by adding a third sine function, now with a period 5/(2T ) and an amplitude 1/5, i.e. 

The process of adding sine functions can be continued giving the following expression  

Summarizing, any signal/(t), measured from time t = -T to t = + T, can be decomposed in a sum of sine and cosine functions as follows  

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e - )/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows  

Equations (40.3) and (40.4) are called the Fourier transform pair. Equation (40.3) represents the transform from the frequency domain back to the time domain, and eq. (40.4) is the forward transform from the time domain to the frequency domain. A closer look at eqs. (40.3) and (40.4) reveals that the forward and backward Fourier transforms are equivalent, except for the sign in the exponent. The backward transform is a summation because the frequency domain is discrete for finite measurement times. However, for infinite measurement times this summation becomes an integral. 

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As mentioned before, the smallest observable frequency (v ,in) in a continuous signal is the reciprocal of the measurement time ( I2T ). Because only those frequencies are considered which exactly fit in the measurement time, all frequencies should be a multiple of namely n/2T with n = -< to -l-oo. As a result the Fourier transform of a continuous signal is discrete in the frequency domain,... 

In summary, the Fourier transform of a continuous signal digitized in 2A/ + 1 data points returns N real Fourier coefficients, N imaginary Fourier coefficients and the average signal, also called the DC term, i.e. in total 2N + 1 points. The relationship between the scales in both domains is shown in Fig. 40.9. 

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