FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS in .NET

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FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
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C. Relationship between the DFT and the Fourier Transform:
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By definition (6.27) the Fourier transform of x [ n ] defined by Eq. (6.91) can be expressed as
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X(fl) =
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(6.97)
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Comparing Eq. (6.97) with Eq. (6.92), we see that
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Thus, X [ k ] corresponds to the sampled X(fl) at the uniformly spaced frequencies f l = k27r/N for integer k .
D. Properties of the D m
Because of the relationship (6.98) between the DFT and the Fourier transform, we would expect their properties to be quite similar, except that the DFT X [ k ] is a function of a discrete variable while the Fourier transform X ( R ) is a function of a continuous n, variable. Note that the DFT variables n and k must be restricted to the range 0 I k < N, the DFT shifts x [ n - no] or X [ k - k , ] imply x [ n -no],,, or X [ k - k,],,, ., where the modulo notation [m],,, means that for some integer i such that
0 For example, if x [ n ] = 6 [ n - 31, then
[mImod~
(6.100)
x [ n - 4],,,
= 6 [ n - 7],,,, = S [ n - 7
+ 61 = 6 [ n - 11
The DFT shift is also known as a circular shift. Basic properties of the DFT are the following:
Time ShifCing:
3. Frequency Shifiing:
4. Conjugation:
where
'*In] ~ ~ * [ - ~ l m o d N denotes the complex conjugate.
CHAP. 61
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
Time Reversal:
6. Duality:
7. Circular Convolution:
where The convolution sum in Eq. (6.108) is known as the circular conuolution of x , [ n ] and 4nI. 8. Multiplication:
where
9. Additional Properties:
When x [ n ] is real, let where x,[n] and xo[n] are the even and odd components of x [ n ] , respectively. Let
x[n] X [ k ] = A [ k ] + j B [ k ] = IX[k]leie[kl
Then
x [- k I m ~ d = X * [ k l ~
I m { X [ k ] ) jB[k] =
( 6 . 110) (6.111a) (6.111b)
k x e [ n ]w R ~ { [X ] )= A [ k ] x o [ n ]- j
From Eq. (6.110) we have
10. Parseval's Relation:
Equation (6.113) is known as Parseual's identity (or Parseual's theorem) for the DFT.
FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
Solved Problems
DISCRETE FOURIER SERIES
6.1. We call a set of sequences ( q k [ n ] orthogonal on an interval [ N , , N , ] if any two signals ) W,[n] and q k [ n ]in the set satisfy the condition
where * denotes the complex conjugate and a exponential sequences
0. Show that the set of complex
is orthogonal on any interval of length N .
From Eq. (1.90) we note that
Applying Eq. (6.116), with a = eik(2"/N), obtain we
since e'k(2"/N'N e jk2" = 1 . Since each of the complex exponentials in the summation in = Eq. (6.117) is periodic with period N, Eq. (6.117) remains valid with a summation carried over any interval of length N. That is,
n=(N)
eiWn/N)n,
k = O , f N , f 2 N , ... otherwise
Now, using Eq. (6.118), we have
where m, k < N. Equation (6.119) shows that the set ( e ~ ~ ( ~ " /= 0). 1 , :. ., N - 1 ) is orthogk ~ ". onal over any interval of length N. Equation (6.114) is the discrete-time counterpart of Eq. (5.95) introduced in Prob. 5.1.
CHAP. 61 FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
Using the orthogonality condition Eq. (6.119), derive Eq. (6.8) for the Fourier coefficients.
Replacing the summation variable k by m in Eq. (6.71, we have
Using Eq. (6.115) with N
= NO, Eq.
(6.120) can be rewritten as
Multiplying both sides of Eq. (6.121) by q t [ n ] and summing over n
to (No - l), we obtain
Interchanging the order of the summation and using Eq. (6.1191, we get
Thus,
Determine the Fourier coefficients for the periodic sequence x [ n ] shown in Fig. 6-7.
From Fig. 6-7 we see that x [ n ] is the periodic extension of (0,1,2,3) with fundamental period No = 4. Thus,
By Eq. (6.8) the discrete-time Fourier coefficients c, are
Note that c,
= c,-,
= cT [Eq. (6.17)].
FOURIER ANALYSIS OF DISCRETE-TIME SlGNALS AND SYSTEMS [CHAP. 6
-4-3-2-1 0
Fig. 6-7
Consider the periodic sequence x [ n ] shown in Fig. 6-8(a). Determine the Fourier coefficients c , and sketch the magnitude spectrum lc,(. From Fig. 6-8(a) we see that the fundamental period of x [ n ] is N o = 10 and R,, =
27r/N,, = ~ / 5 By Eq. ( 6 . 8 ) and using Eq. (1.90),we get .
The magnitude spectrum lckl is plotted in Fig. 6-8(b).
Fig. 6-8
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
Consider a sequence
Sketch x[n]. Find the Fourier coefficients c, of x[n].
The sequence x [ n ]is sketched in Fig. 6-9(a).It is seen that x[n]is the periodic extension of the sequence {1,0,0,O} with period No = 4.
Fig. 6-9
From Eqs. ( 6 . 7 )and (6.8)and Fig. 6-9(a)we have
and since x [ l ]= x[2] = x[3]= 0. The Fourier coefficients of x[n] are sketched in Fig. 6-9(b).
Determine the discrete Fourier series representation for each of the following sequences:
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