Signals and Systems - Electrical Engineering

714 CHAPTER 12: Applications of Discrete-Time Signals and Systems

which requires 16 multiplications (8 if multiplications by 1 are not counted) and 12 additions. Thus,

either 28 or 20, if multiplications by 1 are not counted, multiplications and additions are required.

Since the entries are complex, these are complex additions and multiplications. A complex addition

requires 2 real additions, and a complex multiplication requires 4 real multiplications and 3 real

additions. Indeed, for two complex numbersz=a+jbandv=c+jd,z+v=(a+c)+j(b+c)

andzv=(ac−bd+j(bc+ad)). Thus, the total number of real multiplications is 16×4 and real

additions is 12× 2 + 16 ×3 for a total of 136 operations.

Separating the even- and the odd-numbered entries ofx[n], we have

X[k]=

∑^1

n= 0

x[2n]Wkn 2 +W 4 k

∑^1

n= 0

x[2n+1]Wkn 2

=Y[k]+Wk 4 Z[k] k=0,..., 3

which can be written as

X[k]=Y[k]+W 4 kZ[k]

X[k+2]=Y[k]−W 4 kZ[k] k=0, 1

In matrix form the above equations can be written as

      

X[0]

X[1]

···

X[2]

X[3]

      

=

          

1 0

..

. 1 0

0 1

..

. 0 W^14

··· ··· ··· ··· ···

1 0

..

. − 1 0

0 1

..

. 0 −W^14

          

      

Y[0]

Y[1]

···

Z[0]

Z[1]

      

=A 1











Y[0]

Y[1]

Z[0]

Z[1]











which is in the form indicated by Equation (12.6).

Now we have that

Y[k]=

∑^1

n= 0

x[2n]W 2 kn=x[0]W^02 +x[2]Wk 2

Z[k]=

∑^1

n= 0

x[2n+1]Wkn 2 =x[1]W 20 +x[3]W 2 k k=0, 1