0.8 Appendix b  (Page 5/5)

The first step is to decompose the $2N$ -point DFT into the sum of two N-point DFTs. This is exactly the same operation as is used to start the development of the decimation-in-time (DIT) FFT. Doing this produces the expression

where W _L is defined by the expression $W_L=e^{j\frac{2\pi }{L}}$ . The n -th output is now described by the sum of two N-point DFT taken over real-valued data.

The second step is to use well-known relationships [link] concerning the spectral symmetries of purely real and purely imaginary data. The former has Hermitian symmetry A sequence has Hermitian symmetry if the real parts are symmetrical about the midpoint of the sequence while the imaginary parts are antisymmetrical. while the latter is anti-Hermitian. We exploit this by constructing a new N-point complex sequence $z\left(i\right),0\le i\le N-1$ for each sample instant r according to the rule

This corresponds to packing the $2N$ points of $v(r,p)$ into the real and imaginary parts of an N-point complex-value sequence. Suppose now that we evaluate the DFT of the sequence $z\left(i\right)$ , yielding Z _n . We can break Z _n into the portions, say $Z_n=R_n+j{I}_n$ , where R _n is the real part of the transform and I _n is the imaginary part. The transforms of $v(r,2i),0\le i\le N-1$ , and $v(r,2i+1),0\le i\le N-1$ , are determined by using these Hermitian symmetry properties. In particular, it can be shown that

and

Note that only one N-point DFT plus one additional stage of sums and differences was required to produce both transforms. We can then evaluate [link] to obtain

and

Observe that this computation is essentially the same as one stage of a radix-2, $2N$ -point IFFT. Each desired output $y_n\left(r\right)$ is a bin value of this FFT.

These steps can be summarized follows:

• Compute the $v(r,p)$ according to [link]
• Form the N-point complex-valued sequence $z\left(i\right)$ according to [link]
• Perform the N-point DFT (using an FFT, usually) to obtain Z _n
• Use [link] to obtain the transforms of the two real-valued sequences
• Use [link] and [link] to evaluate [link]

A computational audit of this procedure shows that it requires essentially two more radix-2 stages following the DFT. The first involves only sums and differences while the second, involving the twiddle factors used in a $2N$ -point FFT, requires actual multiplication. A comparison between the multiply-add computation needed for an N-channel FDM-TDM transmultiplexer that accepts complex-valued data at f _s Hz and one that uses the techniques described here and accepts real-valued data at a rate of 2 f _s Hz shows that the only difference is these last two stages. If the transform size is large and/or Q is large, then the computation associated with these two stages may prove negligible, and will almost always be less than that required for a fullband digital tuner. Thus this approach is usually the best if virtually all of the channels in a real-valued signal need to be demultiplexed.

Two other notes in passing:

• The pulse response $h\left(k\right)$ used for weighting the input data must have a duration of 2NQ points for the real-valued case, versus $NQ$ for the complex-valued case.
• The analysis used for real-valued inputs can be combined with that used for obtaining an offset-bin transmultiplexer of the type discussed in Appendix B.2 to obtain an offset-bin design that accepts real-valued data.