FFTs of Real Data

4.7.3 FFTs of Real Data

Frequently the FFT of a real data vector is required. The complex FFT algorithm can still

be used, but is inefficient both in terms of storage and operations. By using symmetries in

4.7. The Fast Fourier Transform 511 the DFT, which correspond to the symmetries noted in the Fourier transform in Table 4.6.1,

better alternatives can be found. Consider the DFT matrix for N = 4 in (4.7.8). Note that the fourth row is the

conjugate of the second row. This is not a coincidence; the conjugate transpose of the DFT matrix F N can be obtained by reversing the order of the last N − 1 rows. Let T N

be the N × N permutation matrix obtained by reversing the last N − 1 columns in the unit matrix

I N . For example,

 1000  T 4 =  0001  

Then it holds that

F N H =F N =T N F N =F N T N .

We first verify that F N =T N F N , by observing that

1 ≤ j ≤ N − 1. Since F N and T N are both symmetric, we also have F H = (T N F N ) T =F N T N .

We say that a vector y ∈ C N is conjugate even if y = T N y , and conjugate odd if y = −T N y . Suppose now that f is real and u = F N f . Then it follows that

u =F N f =T N F N f =T N u,

i.e., u is conjugate even. If a vector u of even length N is conjugate even, this implies that

u j =u N −j , j = 1 : N/2.

In particular, u j is real for j = 0, N/2.

For purely imaginary data g and v = F N g , we have

H v H =F N g = −F N g = −T N F N g = −T N v, i.e., v is conjugate odd. Some other useful symmetry properties are given in Table 4.7.1.

We have proved the first two properties; the others are established similarly and we leave the proofs to the reader; see Problem 4.7.4.

Table 4.7.1. Useful symmetry properties of the DFT.

Data f

Definition

DFT F N f

real conjugate even imaginary

conjugate odd real even

f =T N f real

real odd

f = −T N f imaginary conjugate even f = T N f real conjugate odd f = −T N f imaginary

512 Chapter 4. Interpolation and Approximation We now outline how symmetries can be used to compute the DFTs u = F N f and

v =F N g of two real functions f and g simultaneously. First form the complex function

f + ig and compute its DFT

w =F N (f + ig) = u + iv

by any complex FFT algorithm. Multiplying by T N we have T N w =T N F N (f + ig) = T N (u + iv) = u + iv, where we have used that u and v are conjugate even. Adding and subtracting these two

equations we obtain

w +T N w = (u + u) + i(v + v), w −T N w = (u − u) + i(v − v).

We can now retrieve the two DFTs from

u =F N f =

Re(w + T N w) + i Im(w − T N w) , (4.7.22)

v =F N g =

Im(w + T N w) − i Re(w − T N w) . (4.7.23)

Note that because of the conjugate even property of u and v there is no need to save the entire transforms.

The above scheme is convenient when, as for convolutions, two real transforms are involved. It can also be used to efficiently compute the DFT of a single real function of length N = 2 p . First express this DFT as a combination of the two real FFTs of length N/ 2 corresponding to even and odd numbered data points (as in (4.7.5)). Then apply the procedure above to simultaneously compute these two real FFTs.