NANOSCALE IMAGE PROCESSING - Bio-Inspired and Nanoscale Integrated Computing - page 530

4. This step is the same as step 2, except that the odd-indexed nodes send a
wave with W N k F 2 (k) amplitude on opposite phase. Therefore, the output

node receives Re[F 1 (k) W N k F 2 (k)].

5. This step is similar to step 4, except that the nodes send their ''imaginary''
part. At the end of this stage, X[k+N/2]=F 1 (k) W N k

F 2 (k) is found at the

output node.

As discussed, in this implementation, X[k] values are calculated for k=0, y ,

N/2 1 at one step and for k=N/2, , N 1 in another step. In another

implementation of FFT module, explained in the following, all the N Fourier
coefficients are computed simultaneously.
In the second implementation of FFT module, besides the superposition
property of the waves, we employ another parallel feature of the spin-wave
architectures, i.e., the ability of transmitting multiple waves at different frequen-
cies. The structure of this module is shown in Figure 19.8.
As illustrated, this architecture includes N/2 extra output nodes to compute all
the N outputs simultaneously. The X[k] values for k=0,

y

, N/2 1 are computed
in the first output column, called out1, while X[k] values for k= N/2,

y

, N 1 are
computed in the second column, called out2. These values are computed in the
following steps:

y

1. First the coefficients are multiplied by the inputs in all the nodes (odd-
indexed nodes contain the extra factor of W N k ).
2. At this step, the receiving frequency of the out1 nodes is tuned on f even and
out2 on f odd . The even-indexed input nodes broadcast the ''real'' part of
their data on f even while the odd-indexed nodes broadcast the ''real'' part of
their data on f odd . As a result of the superposition of the spin-waves,
Re[F 1 (k)] is detected in out1 and Re[W N k F 2 (k)] in out2. Since the odd- and

Inputs

Outputs

ω = W N/2

X N -1

x 0 x 1 x 2

X[0] X[N/2]

1 1 1 1

K = 0

x 0 x 1 x 2 X N -1

X[1] X[N/2+1]

3/2 2 N-1

1 ω ω ω

K = 1

X N -1 X[N/2-1] X[N-1]

x 0 x 1 x 2

1

3N/4 − 1/2

ω ( N/2 − 1) 2 ω ( N/2 − 1) N/ 2 ω

K = N / 2 -1

Figure 19.8. Nanoscale spin-wave FFT module (odd-even frequency split method).