DFT MAGNITUDES
The DFT Example 1 results of |X(1)| = 4 and |X(2)| = 2 may puzzle the reader because our input x(n) signal, from Eq. (3-11), had peak amplitudes of 1.0 and 0.5, respectively. There's an important point to keep in mind regarding DFTs defined by Eq. (3-2). When a real input signal contains a sinewave component of peak amplitude Ao with an integral number of cycles over N input samples, the output magnitude of the DFT for that particular sinewave is Mr where
If the DFT input is a complex sinusoid of magnitude Ao (i.e., Aoej2pft) with an integral number of cycles over N samples, the output magnitude of the DFT is Mc where
Equation 3-17'
As stated in relation to Eq. (3-13'), if the DFT input was riding on a DC value equal to Do, the magnitude of the DFT's X(0) output will be DoN.
Looking at the real input case for the 1000 Hz component of Eq. (3-11), Ao = 1 and N = 8, so that Mreal = 1 · 8/2 = 4, as our example shows. Equation (3-17) may not be so important when we're using software or floating-point hardware to perform DFTs, but if we're implementing the DFT with fixed-point hardware, we have to be aware that the output can be as large as N/2 times the peak value of the input. This means that, for real inputs, hardware memory registers must be able to hold values as large as N/2 times the maximum amplitude of the input sample values. We discuss DFT output magnitudes in further detail later in this chapter. The DFT magnitude expressions in Eqs. (3-17) and (3-17') are why we occasionally see the DFT defined in the literature as
Equation 3-18
The 1/N scale factor in Eq. (3-18) makes the amplitudes of X'(m) equal to half the time-domain input sinusoid's peak value at the expense of the additional division by N computation. Thus, hardware or software implementations of the DFT typically use Eq. (3-2) as opposed to Eq. (3-18). Of course, there are always exceptions. There are commercial software packages using
and
Equation 3-18'
for the forward and inverse DFTs. (In Section 3.7, we discuss the meaning and significance of the inverse DFT.) The 1/
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