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Solving this recursion for L(v), we have the following closed-form expression for L ( v ) :

The FFT requires the multiplication of complex numbers:

The output is:

(a) Write out this complex multiplication, and determine how many real multiplies and real adds are

(b) Show that the complex multiplication may also be performed as follows:

tree: [A, B, D, G, E, C, F, H, I]

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The decimation-in-time and decimation-in-frequency FFT algorithms evaluate the DFT of a complexvalued sequence. Show how an N-point FFT program may be used to evaluate the N-point DFT of two real-valued sequences.

As we saw in Prob. 6.18. the DFTs of two real-valued sequences may be found from one N-point DFT as follows. First, we form the N-point complex sequence

The program creates the binary tree shown in Figure 11.21 and then indirectly invokes its toString() method that it inherits from the AbstractCollections class. Figure 11.21 shows two views of the same tree. The larger view shows all the details, representing each object reference with an arrow.

After finding the N-point DFT of .r(n). we extract X I ( k )and X z ( k ) from X ( k ) by exploiting the symmetry of the DFT. Specifically.

which is the conjugate antisymmetric part of X ( k ) .

CHAP. 11]

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CHAP. 11]

-k ) ) ~ ]

By extending the AbstractCollection class, the BinaryTree class automatically inherits these methods that are defined by using the iterator() and size() methods:

Determine how a 2N-point DFT of a real-valued sequence may be computed using an N-point FFT algorithm.

By extending the AbstractCollection class, the BinaryTree class automatically inherits these methods that are defined by using the iterator() and size() methods:

Let g ( n ) be a real-valued sequence of length 2N. From this sequence. we may form two real-valued sequences of length N as follows:

isEmpty() contains(Object object) toArray() toArray(Object[] objects) toString() add(Object object) addAll(Collection collection) clear() containsAll(Collection collection) remove(Object object) removeAll(Collection collection) retainAll(Collection collection) However, the mutating methods will throw an UnsupportedOperationException because they invoke other methods that are not implemented, namely the add() and the Iterator.remove() methods. public public public public public public public public public public public public boolean boolean Object[] Object[] String boolean boolean void boolean boolean boolean boolean

Computing the N-point DFT of . r ( n ) . we niay then extract the N-point DFTs of x , ( n ) and x 2 ( n ) as follows (see Prob. 7 . 7 ) :

x ~ ( k= i [ ~ ( k ) X * ( ( N - k ) ) ~ ] ) x ~ k = ; [ x ( k ) - x * ( ( N - k ) ) NI )

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