76 DISCRETE-SIGNAL ANALYSIS AND DESIGN WINDOWING REFERENCES Harris, F. J., 1978, On the use of windows for harmonic analysis with the Fourier transform, Proc. IEEE ,Jan. Oppenheim, A. V., and R. W., Schafer, 1975, Digital Signal Processing, McGraw-Hill, New York. 5 Multiplication and Convolution Multiplication and convolution are very important operations in discrete sequence operations in the time domain and the frequency domain. We will Þnd that there is an interesting and elegant relationship between mul- tiplication and convolution that is useful in problem solving. MULTIPLICATION For the kinds of discrete time x(n) or frequencies X(k)ofinterestinthis book, there are two types of multiplication. The (n) and (k ) values are integers from 0 to N −1. The X (k) values to be multiplied are phasors that have amplitude, frequency, and phase attributes, and the x (n) values have amplitude and time attributes. The Mathcad program sorts it all out. Each sequence is assumed by the software to be one realization of an inÞnite, steady-state repetition, with all of the signiÞcant information available in a single two-sided (n)or(k) sequence, as explained previously and mentioned here again for emphasis. Discrete-Signal Analysis and Design, By William E. Sabin Copyright 2008 John Wiley & Sons, Inc. 77 78 DISCRETE-SIGNAL ANALYSIS AND DESIGN Sequence Multiplication One type of multiplication is the distributed sequence multiplication seen in Eq. (4-2) and repeated here: z(n) = x(n) y(n),0≤ n<N−1 (5-1) Each element of z (n) is the product of each element of x (n) and the corresponding element of y(n). Frequently, x(n) is a “weighting factor” for the y(n) value. For example, x(n) can be a window function that modiÞes a signal waveform y(n). Chapter 4 showed some examples that will not be repeated here. The values x (n)andy(n) may in turn be functions of one or more parameters of (n) at each value of (n), which is “grunt work” for the computer. We are often interested in the sum z(n) over the range 0 to N −1 as the sum of the product of each x(n) and each y(n). Also, the time average or mean-square value of the sum or other statistics is important. And of course, we are especially interested in Z (k), the spectrum of z (n), Y (k), the spectrum of y(n), and X (k), the spectrum of x(n). Figure 5-1 is another example of frequency conversion by using this kind of multiplication. A time sequence x (n) at a frequency k =4and a time sequence y(n) at frequency k =24 are multiplied term-by-term to get the time sequence for the product. The DFT then Þnds the two-sided phasor spectrum. The one-sided spectrum is found by adding the phasors at 108 and 20 to get the positive cosine at 20, and adding the 100 and 28 phasors to get the negative cosine term at 28. See Fig. 2-2a to conÞrm these results, and note the agreement with the equation for z (i ) in Fig. 5-1. As we said before, the input frequencies 24 and 4 disappear, but if one input amplitude is held constant, the product is linear with respect to variations in the other input amplitude. Polynomial Multiplication The other kind of sequence multiplication is polynomial multiplication, which uses the distributive and associative properties of algebra. An example is shown in Eq. (5-2). MULTIPLICATION AND CONVOLUTION 79 x(i) := sin 2·π· i N ·4 y(i) := sin 2·π· i N ·24 0 20 40 60 80 100 120 −2 0 2 x(i) 0 20 40 60 80 100 120 −2 0 2 y(i) z(i) := (x(i)·y(i)) := z(i) 1 2 cos 2·π·(24 − 4)· 2·π·(24 + 4)· i N 1 2 cos i N 0 20 40 60 80 100 120 −2 0 2 i i i Z(k) := Z(k) 1 N z(i)·exp −j·2·π· i N ·k · 0 26 52 78 104 130 −0.5 −0.25 0 0.25 0.5 k N = 128 i = 0,1 N−1 − ∑ N−1 i = 0 Figure 5-1 Frequency conversion through term-by-term multiplication of two time sequences. 80 DISCRETE-SIGNAL ANALYSIS AND DESIGN z(x,y) = (x 1 + x 2 +···+x α )(y 1 + y 2 + y 3 +···+y β ) = y x z(x) z(y) z(i) = x(i)(y 1 + y 2 + y 3 +···+y β ) = x(i) j y(j) (5-2) Each term of the x sequence is multiplied by the sum of the terms in the (y) sequence to produce each term in the (z ) sequence, which then has α terms. Or, each term of the (y) sequence is multiplied by the sum of the terms in the (x) sequence to get β terms. Or, each term in the Þrst sequence is multiplied by each term in the second sequence and the partial products are added. In all of these ways, the (z ) sequence is the polyno- mial product of the (x ) and (y) sequences. The sum of z(i) is the “energy” in (z ). This, divided by the “time” duration, is the “average power” in z (i ). If the sum of (x )orthesumof(y) equals zero, the product is zero, as Eq. (5-2) clearly indicates. For certain parts of the range of x (i)and y(j ) the product can usually be nonzero. In the familiar arithmetic multi- plication, the (x ) and (y) terms are “weighted” in descending powers of 10 to get the correct answer: for example, 8734 · 4356 = (8000 + 700 +30 + 4) ·(4000 + 300 +50 + 6) = 8000 · 4000 +8000 · 300 + 8000 ·50 + 8000 ·6 +700 · 4000 +700 · 300 +700 · 50 +700 · 6 +30 · 4000 +30 · 300 +30 · 50 +30 · 6 +4 · 4000 +4 · 300 +4 · 50 +4 · 6 = 38, 045, 304 (5-3) Polynomial multiplication A(x ) B(y) is widely used, including in topics in this and later chapters. It shows how each item in sequence A(x) affects a set of items in sequence B(y). It is equivalent to a double integration or double summation such as we might use to calculate the area of a two-dimensional Þgure. Figure 5-2 is a simple example. More complicated geometries require that the operation be performed in segments and the partial results combined. . previously and mentioned here again for emphasis. Discrete-Signal Analysis and Design, By William E. Sabin Copyright 2008 John Wiley & Sons, Inc. 77 78 DISCRETE-SIGNAL ANALYSIS AND DESIGN Sequence. 76 DISCRETE-SIGNAL ANALYSIS AND DESIGN WINDOWING REFERENCES Harris, F. J., 1978, On the use of windows for harmonic analysis with the Fourier transform, Proc. IEEE ,Jan. Oppenheim, A. V., and. The (n) and (k ) values are integers from 0 to N −1. The X (k) values to be multiplied are phasors that have amplitude, frequency, and phase attributes, and the x (n) values have amplitude and time