3.2. DFT SYMMETRY
Looking at Figure 34(a)
again, there is an obvious symmetry in the DFT results. Although
the standard DFT is designed to accept complex input sequences,
most physical DFT inputs (such as digitized values of some
continuous signal) are referred to as real, that is, real inputs have nonzero
real sample values, and the imaginary sample values are assumed to
be zero. When the input sequence x(n) is
real, as it will be for all of our examples, the complex DFT
outputs for m = 1 to m = (N/2) – 1 are redundant with
frequency output values for m >
(N/2). The mth DFT output will have the same
magnitude as the (N–m)th DFT output. The phase angle of the
DFT's mth output is the negative
of the phase angle of the (N–m)th DFT output. So the mth and (N–m)th outputs are related by the
following:
We can state that when the DFT input sequence is
real, X(m) is the complex conjugate of X(N–m), or
Equation 314'
where the superscript ^{*} symbol
denotes conjugation.
In our example above, notice in Figures 34(b)
and 34(d) that
X(5), X(6), and X(7) are the complex conjugates of X(3), X(2), and X(1), respectively. Like the DFT's
magnitude symmetry, the real part of X(m)
has what is called even symmetry,
as shown in Figure 34(c),
while the DFT's imaginary part has odd
symmetry, as shown in Figure 34(d).
This relationship is what is meant when the DFT is called conjugate
symmetric in the literature. It means that, if we perform an Npoint DFT on a real input sequence,
we'll get N separate complex DFT
output terms, but only the first N/2+1 terms are independent. So to
obtain the DFT of x(n), we need only compute the first N/2+1 values of X(m)
where 0 m (N/2); the X(N/2+1) to X(N–1) DFT output terms provide no
additional information about the spectrum of the real sequence
x(n).
Although Eqs. (32) and (33) are
equivalent, expressing the DFT in the exponential form of Eq.
(32) has a terrific advantage over the form of Eq.
(33). Not only does Eq. (32) save pen and
paper, Eq. (32)'s
exponentials are much easier to manipulate when we're trying to
analyze DFT relationships. Using Eq. (32), products of
terms become the addition of exponents and, with due respect to
Euler, we don't have all those trigonometric relationships to
memorize. Let's demonstrate this by proving Eq. (314) to show the symmetry of the DFT of
real input sequences. Substituting N–m for m
in Eq.
(32), we get the expression for the (N–m)th component of the DFT:
Equation 315
^{} Using our notation, the complex conjugate of
x = a + jb
is defined as x^{*} =
a – jb; that is, we merely change the sign
of the imaginary part of x. In an
equivalent form, if x = e^{j}^{ø}, then
x* = e^{–}^{j}^{ø}.
Because e^{–}^{j}^{2}^{p}^{n} = cos(2pn)
–jsin(2pn) = 1 for
all integer values of n,
Equation 315'
We see that X(N–m) in Eq. (315') is merely X(m) in
Eq.
(32) with the sign reversed on X(m)'s
exponent—and that's the definition of the complex conjugate.
This is illustrated by the DFT output phaseangle plot in Figure 34(b)
for our DFT Example 1. Try deriving Eq. (315') using the cosines and sines of Eq.
(33), and you'll see why the exponential form of the DFT is so
convenient for analytical purposes.
There's an additional symmetry property of the
DFT that deserves mention at this point. In practice, we're
occasionally required to determine the DFT of real input functions
where the input index n is defined
over both positive and negative values. If that real input function
is even, then X(m) is always real and even; that is, if
the real x(n) = x(–n), then, X_{real}(m) is in general nonzero and X_{imag}(m) is zero. Conversely, if the real
input function is odd, x(n) = –x(–n), then X_{real}(m) is always zero and X_{imag}(m) is, in general, nonzero. This
characteristic of input function symmetry is a property that the
DFT shares with the continuous Fourier transform, and (don't worry)
we'll cover specific examples of it later in Section
3.13 and in Chapter 5.
