信号与系统双语课件chapter1.2

2 sinusoidal signals
When βis purely imaginary, then
x[n] e j0n
As in the continuous-time case, this signal is closely related to the sinusoidal signal
e
j0t
cos0t j sin 0t
Similarly , the sinusoidal signals can be written in terms of periodic complex exponential with the same fundamental period:
A j j 0 t A j j 0 t A cos( 0t ) e e e e 2 2
1.3.3 Periodicity properties of discrete-Time complex Exponentials
1.Idential signal for values of 0 separated by multiple of 2
e
j (0 2 ) n
e
j 2n
(3) General complex exponential signal If C and a are complex and we write C in polar form and a in rectangular form:
C C e j a r j0
Then
cos(0t ) j sin(0t )
x(t ) Ce at
Where C and a are complex numbers. Depending upon the values of these parameters, the complex exponential can exhibit several different characteristics.
1.3.2 Discrete-Time complex Exponential and sinusoidal Signals
The discrete-time complex exponential signal is of the form
x[n] C n
Where C and are complex number. This could alternatively be expressed in the form
1.2.3 Even and odd signals
1 Even signal
x(t ) x(t ) CT signal x(n) x(n) DT signal
Is y t 2 a evensignal or not?
2 Odd signal
x(t ) x(t ) CT signal x(n) x(n) DT signal
2m 2. Periodic only if 0 for some integers N>0 and m. N
j n In order for the signal e 0 to be periodic with period N>0,
we must have
e
j0 ( n N )
periodic CT signals
From the figure, we can deduce that if x(t) is periodic with period T, then x(t)=x(t+mT) for all t and for any integer m. Thus,x(t) is also periodic with period 2T,3T,4T…. The fundamental period T of 0 x(t) is the smallest positive value of T.
For a discrete-time signal, if
x[n] x[n N ]
for any value of n and N is a positive integer. We say this signal is periodic with period N. x[n] is also periodic with period 2N,3N,4N…. The fundamental period N 0 of x[n] is the smallest positive value of N.
e
j0 n
e
j0 n
So ,we see that the exponential at frequency 0 2 is the same as that at frequency 0 .therefore ,in considering discretetime complex exponentials, we need only consider a frequency interval of length 2 in which to choose 0 .
1.3 Exponential and sinusoidal Signals
1.3.1 Continuous-Time complex Exponential and sinusoidal Signals
The continuous-time complex exponential signal is of the form
Is y n a even signal or odd?
An important fact is that any signal can be broken into a sum of two signals ,one of which is even and one of which is odd. So
Because:
Ae
j (0t )
Asin(0t ) jA cos(0t )
We can express a sinusoidal signal in terms of complex exponential with as:
Asin(0t ) A Im{ e j (0t ) } A cos(0t ) A Re{e j (0t ) }
x(t ) Ceat C e j e( r j0 )t C ert e j (0t )
And
x(t ) Ceat C ert cos(0t ) j C ert sin(0t )
For r >0,the real and imaginary parts of a complex exponential correspond to sinusoidal signals multiplied by growing exponential in figure a, and for r<0,they correspond to sinusoidal signals multiplied by decaying exponential in figure b. The dashed lines in the figure correspond to the functions C ert .
1 Real exponential signals
If C and a are real, in which case x(t) is called a Real exponential signals
x(t ) Ce
at
a>0
a<0
2 Periodic complex exponential and sinusoidal signals
n
n
n
For 1 ,they correspond to sinusoidal signals multiplied by growing exponential .
For 1 ,they correspond to sinusoidal signals multiplied by decaying exponential.
e
j0 n
Or equivalently
e
j0 N
1
So 0 N must be a multiple of 2 .There must be an integer m
1.2.2 Periodic signals
A periodic continuous-time signal x(t) has the property that there is a positive value of T for which: x(t ) x(t T ) for all value of t. In other words, a periodic signal has the property that it is unchanged by a time shift of T. in this case, we say that x(t) is periodic with period T.
x[n] Ce n
Where
e

1 Real exponential signals
If C and signals
α are real, in which case x[n] is called a Real
exponential
Thereal exponentia l signal x[n] C n : (a) 1; (b)0 1; (c) 1 0; (d ) 1.
(1) If a is to be purely imaginary
x(t ) e j0t
An important property of this signal is that it is periodic. So
e j0t e j来自百度文库 (t T )
If 0 0,the fundamental period T0 of x(t) – that is the smallest positive value of T –is:
3 General complex exponential signal
If we write C and αin polar form:
C C e j
and
e j
0
Then
x[n] C C cos( 0 n ) j C sin(0 n )
x[n] A cos(0 n )
As before, Euler’s relation allow us to relate complex exponential and sinusoids:
e
and
j0n
cos0n j sin 0n
A j j 0 n A j j 0 n A cos( 0 n ) e e e e 2 2
x(t ) Ev{x(t )} od{x(t )}
1 Ev{x(t )} [ x(t ) x(t )] 2 1 Od {x(t )} [ x(t ) x(t )] 2
And
For y 2t t 2 , how to brokenit intoa sum of two signals?
T0
2
0
(2) Sinusoidal signal
x(t ) A cos(0t )
By using Euler’s relation, the complex exponential can be written in terms of sinusoidal signals with the same fundamental period: