信号与系统双语课件chapter3.3

Since the product x(t)y(t) is also periodic with period T, we can expand it in a Fourier series with Fourier series coefficients hk expressed in terms of those for x(t) and y(t). The result is
The Dirichlet conditions are as follow: Condition 1. over any period, x(t) must be absolutely integrable, that is x(t ) dt
T
Condition 2. In any finite interval of time, x(t) is of bounded variation; that is, there are no more than a finite number of maxima and minima during any single period of the signal. Condition 3. In any finite interval of time, there are only a finite number of discontinuities. If a signal satisfy the Dirichlet conditions, it can be represented by Fourier series.
FS x(t ) ak
to signify the pairing of the periodic signal with its Fourier series coefficients. That means the Fourier series coefficients of x(t) are {ak}
3.5.2 Time shifting
If
FS x(t ) ak
Then
FS x(t t0 ) e jk0t0 ak e jk (2 / T )t0 ak
One consequence of this property is that, when a periodic signal is shifted in time, the magnitudes of its Fourier series coefficients remain unaltered. That is,
x(t ) y(t ) hk
FS
l
a b

l k l
3.5.6 Conjugation and conjugate symmetry
If
FS x(t ) ak
FS x (t ) a k
x (t ) is the complex conjugation of the periodic signal x(t).
a k is the complex conjugation of the periodic signal a-k.
3.5.7 Parseval’s relation for continuoustime periodic signals
The parseval’s relation :the total average power in the a periodic signal equals the sum of the average power in all of its harmonic components
FS x(t ) ak FS y(t ) bk
Then the Fourier series coefficients ck of the linear combination of x(t) and y(t), z(t) =Ax(t)+By(t), are given by the same linear combination of the Fourier series coefficients for ak and bk.
★ An consequence of this property is that if x(t) is even, then its Fourier series coefficients are also even. Similarly , if x(t) is odd, the its Fourier series coefficients are also odd.
For a signal
x[n] e
j0n
It is a periodic signal with fundamental frequency 0 2 / N and fundamental period N .
Associated with this signal is the set of harmonically related complex exponentials
ak bk
3.5.3 Time reversal
If
FS x(t ) ak FS x(t ) ak
Then
Time reversal applied to a continuous-time signal results in a time reversal of the corresponding sequence of Fourier series coefficients.
3.5.1 Linearity
Let x(t) and y(t) denote two periodic signals with period T and which have Fourier series coefficients denote by ak and bk, respectively.
3.5 Properties of continuous-time Fourier series
Fourier series representations possess a number of important properties that are useful for developing conceptual insights into such representations, and they can also help to reduce the complexity of the evaluation of Fourier series of many signals. Throughout the following discussion of the selected properties, we will use the notation
2 1 x(t ) dt ak T T k 2
3.6 Fourier series representation of dicrete-time periodic signals
3.6.1 Linear combination of harmonically related complex exponential
1 1 jk0t ak x(t )e dt T T T

T
x(t )e jk ( 2 / T )t dt
We denote integration over any interval of length T by

T
3.4 Convergence of the Fourier series
review
If a signal x(t) has a Fourier series representation in the form of
x(t )
k
ak e

jk0t

k
jk ( 2 / T ) t a e k

Then the coefficients are given by
FS z(t ) Ax(t ) By(t ) ck Aak Bbk
The period of z(t) is also T.
Representation of the periodic signal z(t) as a weighted sum of periodic square waves: z(t) = (3/2)x(t) + 1/2y(t). (a) z(t). (b) x(t). (c) y(t).
k [n] e jk n e jk (2 / N ) n , k 0,1,2,....
0
There are only N distinct signals in the this set. why? From section 1.3.3, we know a fact that discrete-time complex exponentials which differ in frequency by a multiple of 2πare identical. So :
k [n] e jk n e jk (2 / N ) n , k 0,1,2,....
0
Each of these signals has a fundamental frequency that is a multiple of0 and therefore, each is periodic with period T.
k [n] k rN [n]
That is ,when k is changed by any integer multiple of N, the identical sequence is generated.
Now we wish to consider the representation of more generally periodic sequences in terms of linear combinations of the sequences k [n] ,such a linear combination has the form