SYMMETRIES

Next we examine odd/even symmetries to see how they are affected in Fourier transform. The even part e_t of a signal b_t is defined as  

$$e_t \eq {b_t + b_{-t} \over 2}$$ (12)

The odd part is  

$$o_t \eq {b_t - b_{-t} \over 2}$$ (13)

By adding (12) and (13), we see that a function is the sum of its even and odd parts:  

$$b_t \eq e_t + o_t$$ (14)

Consider a simple, real, even signal such as (b_-1, b₀, b₁) = (1, 0, 1). Its transform $Z + 1/Z = e^{i\omega}+e^{-i\omega} = 2\cos \omega$is an even function of $\omega$, since $\cos \omega = \cos (-\omega)$.

Consider the real, odd signal (b_-1, b₀, b₁) = (-1, 0, 1). Its transform $Z - 1/Z = 2i\sin\omega$ is imaginary and odd, since $\sin \omega = -\sin(-\omega)$.

Likewise, the transform of the imaginary even function (i, 0, i) is the imaginary even function $i2 \cos \omega$.Finally, the transform of the imaginary odd function (-i, 0, i) is real and odd.

Let r and i refer to real and imaginary, e and o to even and odd, and lower-case and upper-case letters to time and frequency functions. A summary of the symmetries of Fourier transform is shown in Figure 6.

 

reRE Figure 5 Odd functions swap real and imaginary. Even functions do not get mixed up with complex numbers.

More elaborate signals can be made by adding together the three-point functions we have considered. Since sums of even functions are even, and so on, the diagram in Figure 6 applies to all signals. An arbitrary signal is made from these four parts only, i.e., the function has the form $b_t = ( {\rm re} + {\rm ro})_t + i ( {\rm ie} + {\rm io})_t$.On transformation of b_t, each of the four individual parts transforms according to the table.

Most ``industry standard'' methods of Fourier transform set the zero frequency as the first element in the vector array holding the transformed signal, as implied by equation (3). This is a little inconvenient, as we saw a few pages back. The Nyquist frequency is then the first point past the middle of the even-length array, and the negative frequencies lie beyond. Figure 7 shows an example of an even function as it is customarily stored.

 

even
Figure 6 Even functions as customarily stored by ``industry standard'' FT programs.