In practical problems this sequence is assumed to have “useful” amplitude only between two speciÞc limits, mmin and mmax.. Sequence hm refers to a “system function,” also a time-domain s
Trang 1Δy Xmin
Xmax Ymin Ymax
x
Δx
y
Δy Area = z(x,y) =
Σ Δy
Figure 5-2 Example of polynomial multiplication using double
summa-tion to Þnd the area of a Þgure
CONVOLUTION
Convolution is a valuable tool for the analysis and design of communi-cations systems and in many other engineering and scientiÞc activities Equation (5-4) is the basic equation for discrete-time convolution
y(n) = x(m) ∗ h(m) =
+∞
m=−∞
[x(m) h(n − m)] (5-4)
where ∗ is the convolution operator and y(n), x(m), and h(m) can all be the complex-valued discrete-time sequences I and jQ that we considered carefully in Chapter 1 Note that x(m), h(m), and y(n) are in the time domain, but they can also be complex Y (k), X (k), and H (k) in the
fre-quency domain with magnitude and phase attributes Also, all three can have different amplitude scale factors, on the same graph or on separate graphs We focus initially on the time domain
Equation (5-4) appears to be simple enough, but actually needs some careful study and practice to develop insight and to assure correct answers
Trang 2Sequence x(m) is a “signal” input time-domain sequence that extends in
“time” from m = − ∞ to m = + ∞ In practical problems this sequence
is assumed to have “useful” amplitude only between two speciÞc limits,
m(min) and m(max) Sequence h(m) refers to a “system function,” also a time-domain sequence that is assumed to have useful limits from m(min) to m(max), which may not be the same limits as the limits for x(m) Sequence h(n − m) is h(m) that has had two operations imposed: 1: h(m) has been
“ßipped” in time and becomes h( − m), and 2: h( − m) has been shifted to the right (n) places, starting from an initial value of (n) determined by the nature of the problem, whose value is now h(n − m) The expression “fold
and slide” is widely used to describe this two-part operation
One reason for the fold and slide of h(m) to h(n − m) is that we want the leading edge in time of h(m) (the response) to encounter the leading edge in time of x(m) (the excitation) as we start the sliding operation (increasing n) from starting (n) to Þnal (n) This retains the time
coordi-nation between the two sequences Another reason is that this procedure leads to a valuable concept that we will demonstrate later in this chapter
The term h(n − m) is often referred to as an impulse response Each impulse x(m) applied to a network is processed by some impulse response function h(n − m) to produce an output impulse y(n), which is the “value”
of the convolution for that (n) Note the summation sign in Eq (5-4) Refer to Fig 5-3 In part (a), h(m) and x(m) are at Þrst located side by side This is a convenient starting point In part (b), h(m) is ßipped and piv-oted about the m = 0 point and becomes h( − m) At each positive increment
of (n), h(n − m) is advanced one position to the right We now calculate the product of x(m) and h(n − m) at each value of (m), and add these products from x(min) to x(max) to get y(n), the convolution value of x(m) and h(m)
at that value of (n) We continue this shift–multiply–add procedure until the
value of the sum of products (the convolution value) becomes negligible
Make sure that the data space (0 to N − 1) is enough that the total convolution sum can be completed with no signiÞcant loss of data If the
x(m) sequence has Lx values and the side-by-side h(m) sequence has Lh values, the y(n) sequence should have at least Lx + Lh + 1 values prior to
the beginning of overlap The convolution operation creates a smoothing
and stretching operation on the data y(n) which is not obvious in Figs.
5-3 and 5-4 but is more visible if we look ahead to Fig 5-7
The convolution sequence that has been generated has properties that will be discussed subsequently This procedure should be compared to the
Trang 30
0
0
0
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−3
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−2
−2
0
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m
m
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m
m
−1
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−1
−1
−1
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2 0 0 0 0
0
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6 6
6 6
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9
h(0 −m)
h(m)
h(5 −m)
h(8 −m)
h(11 −m)
x(m) x(m)
x(m)
x(m)
x(m)
(a)
(b)
(c)
(d )
(e)
y(0) = 0
y(5) = 6 x 2 + 7 x 0
= 12
y(8) =
7 x 6 + 8 x 4 + 9 x 2
= 92
y(11) = 0
9 9
9
Figure 5-3 Various stages of discrete convolution.
Trang 4(b)
(c)
(d)
(e)
y(0) = 0
y(8) =
9 x 2 + 8 x 4 + 7 x 6
= 92
y(5) =
6 x 2 + 7 x 0 = 12
y(11) = 0
0 1 2 3 4 5 6 7
−7 −6 −4 −3
0
5
10
m
−1
2 0 4
6 6 7
8 9
x(m)
h(m)
−9
0 1 2 3 4 5 6 7
−7 −6 −4 −3
0
5
10
m
−1
2 0 4 6 6
7 8 9
x(m) h(0-m)
−9
0 1 2 3 4 5 6 7
−7 −6 −4 −3
0
5
10
m
−1
2 0 4 6 6 7 8 9
x(m) h(5−m)
−9
0 1 2 3 4 5 6 7
−7 −6 −4 −3
0
5
10
m
−1
2 0 4
67 6 8 9
x(m)
h(8−m)
−9
0 1 2 3 4 5 6 7
−7 −6 −4 −3
0
5
10
m
−1
2 0 4
8
x(m)
h(11 −m)
−9
9
Figure 5-4 Altered stages of discrete convolution.
Trang 5convolution of continuous functions, in which certain discontinuities in the functions and their overlaps require integrations over adjacent areas which are then tacked together end-to-end to get the Þnal result Our example sequences will not have this problem because of their discrete nature, although this may not always be true
This process is an example of the polynomial multiplication shown in
Eq (5-2) That is, at a particular value of (n), the product of each x(m) and the corresponding h(n − m) is formed These partial products are summed over the range m(min) to m(max) to get the convolution value y(n) This process is repeated for each value of (n).
To illustrate this discussion more numerically, we see again in Fig 5-3
the convolution procedure for the two discrete sequences shown in part (a) The m= 0 location is chosen arbitrarily as shown, but in principle
it can be anywhere between− ∞ and + ∞ In agreement with our previ-ous assumptions, the process shown is signiÞcant between 0≤ n ≤ N − 1, where N is chosen to suit the problem A convenient approach is to start the h(m) sequence at a point labeled “0”.
In Fig 5-3b the h(m) sequence is reversed (or folded) and pivoted around the zero location, shown at the left end of h(m), and becomes h( − m) That is, each m (positive location) becomes m (negative loca-tion) After pivoting, h( − m) can be moved three places to the right, where it becomes h(3 − m), just one “bin” away from x(m).
In part (c) the h(3 − m) sequence slides to the right two places (n = 5) and overlaps the Þrst two positions of the x(m) sequence The value of the convolution for n= 5 is 6·2 + 7·0 = 12
In part (d) the value of (n) is 8 and the convolution sum is 7·6 + 8·4 +
9·2 = 92 At n = 11 the overlap of h(n − m) and x(m) is zero and the convolution sum for n= 11 is zero
In Fig 5-4 x(m) and h(m) are interchanged; x(m) is response and h(m) is the signal, with no differences in the convolution values, which
is an interesting facet of convolution that is the “commutative” property
We see also that convolution involves two time-domain sequences or two frequency-domain sequences and the convolution value (sum of products) has the dimension of energy
Discrete Circular Convolution
In the sequences used in convolution, any empty integer locations are Þlled
with zeros by Mathcad The combined length L of the two sequences x(m)