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The second way that the DFT may be used to perform linear convolution is to use the overlap-save method. This method takes advantage of the fact that the aliasing that occurs in circular convolution only affects a portion of the sequence. For example, if .r(n) and h ( n ) are finite-length sequences of lengths L and N, respectively, the linear convolution y ( n ) is a finite-length sequence of' lengths N L - I . Therefore, assuming that N > L, if we perform an N-point circular convolution of x ( n ) with h ( n ) ,

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The Comparable interface requires the compareTo() method, as specified at line 2. The purpose of the method is to indicate whether the implicit argument is less than, equal to, or greater than the explicit argument. The indicator is the sign of the returned integer: Negative indicates that this < object, zero indicates that this = object, and positive indicates that this > object. In the case of ratios a/b and c/d, we can tell whether a/b < c/d by cross-multiplying and checking whether ad < bc. Since that case should be indicated by returning a negative number, we can simply return ad bc. Indeed, the value of that expression will be negative, zero, or positive, when a/b < c/d, a/b = c/d, or a/b > c/d, respectively. However, that arithmetic trick works only if b > 0 and d > 0. To ensure that the denominators are not negative, we employ the static utility method defined at line 14. It ensures that the denominator of its argument is not negative.

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These are faster converging implementation of the sine and cosine functions:

Because y(n N ) is the only term that is aliased into the interval 0 5 n 5 N - 1, and because y ( n N) only overlaps the first L - 1 values of y ( n ) , the remaining values in the circular convolution will not be aliased. In other words, the first L - 1 values of the circular convolution are not equal to the linear convolution, whereas the last M = N - L I values are the same (see Fig. 6-5). Thus, with the appropriate partitioning of the input sequence x ( n ) . linear convolution may be performed by piecing together circular convolutions. The procedure is as follows:

Perform the N-point circular convolution of x l ( n ) with lz(n) by forming the product H ( k ) X I(k) and then finding the inverse DFT, y l ( n ) . The first L - 1 values of the circular convolution are aliased, and the last

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Figure 1.7 illustrates how the implementation of an interface is represented in UML. The association symbol is a dashed arrow pointing to the interface. The diagram for the interface marks its name with the stereotype interface . Note that an interface rectangle has only two parts, while a class rectangle has three. This is because interfaces have no fields. Figure 1.7 also illustrates how to represent package membership. The Comparable interface is part of the java.lang package, as indicated by the tab above the interface symbol. POLYMORPHISM

public static double s(double x) { if (-0.005 < x && x < 0.005) { return x - x*x*x/6 + x*x*x*x*x/120; } return 2*s(x/2)*c(x/2); // recursion } public static double c(double x) { if (-0.005 < x && x < 0.005) { return 1.0 - x*x/2 + x*x*x*x*x/24; } return 1 - 2*s(x/2)*s(x/2); // recursion } // basis

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N - L 1 values correspond to the linear. convolution of x ( n ) with h ( n ) . Due to the zero padding at the start of x l ( n ) , these last N - L I values are the first N -- L 1 values of y ( n ) :

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