國立臺灣海洋大學資訊工程學系 訊號與系統 TWP(2009)
§ Fundamental Concepts
l Signals: is a real-valued, function of the time variable
represented : mathematical function ex.
a set of sample values
frequency contents (frequency spectrum)
● Signal Processing : information extraction
reconstruction of signals (estimation/filtering)
● Systems is an interconnection of components with access ports.
representation : mathematical model (idealized)
two basic types : I/O representation : differential eq. / difference eq.
convolution model
Fourier transform representation
transfer function
State model
● Continuous-Time Signals
: when time variable takes its values from the set of the real number
● Step function
unit-step function
= 1,
0,
= ,
0,
● Ramp function
unit-ramp function
= ,
0,
● Impulse
unit-impulse function
= 0,
for any real number
, all t except t=0.
● Periodic signals
A continuous-time signal is periodic with period T if , for all t.
if , are periodic signals, is periodic?
Examples: , is periodic ?
, is periodic ?
● Time-Shift signals
If , 指shifted to right by seconds
指shifted to left by seconds
example:
shift property of impulse:
proof: f(t) is continuous at t1,
example:
● Scaling : , a is a positive real number;
, k is a positive integer
● Flip(reversal):
Example of transformation of independent variable
, what is the waveform of y(t)?
● Continuous and Piecewise-Continuous Signals
is discontinuous at if (i.e. jump at ).
is continuous at if .
If a continuous-time signal is continuous at all points , then is said to be a continuous signal.
examples:
rectangular pulse function pτ(t) triangular pulse function vτ(t)
Try to sketch p5(t-3) and v6(t+4).
is piece wise continuous if it is continuous at all points except at a finite or countably infinite collection of points .
examples: pulse train
● Derivative of a continuous-time signal
Generalized derivative of (if discontinuous at )
examples:
Continuous at all t except t=0,1
Ordinary derivative of
Generalized derivative of
● Signals defined interval by interval
From previous example :
● Using MATLAB to plot continuous-time signals
Simply to define t and x as follows, then use plot command plot(t,x)
t=[-2 -1 0 0 1 1 2 3 4 5 6];
x=[0 0 0 1 3 1 1 0 0 0 0];
plot(t,x);
%
% Example of a continuous-time signal
% (200~400 points for plotting are good choice)
t=0:0.1:30;
x = exp(-.1*t).*sin(2/3*t);
plot(t,x)
grid
axis([0 30 -1 1]);
ylabel('x(t)')
xlabel('Time (sec)')
title('Continuous-time Signal')
%
% Example of a continuous-time signal
% (different resolution)
t=0:2:30;
x = exp(-.1*t).*sin(2/3*t);
plot(t,x)
grid
axis([0 30 -1 1]);
ylabel('x(t)')
xlabel('Time (sec)')
title('Continuous-time Signal')
● Discrete-Time Signals
A signal that is a function of the discrete-time variable .
: if time variable takes on only the discrete value for some range of
integer values of ( )
● Sampling
● Step function
unit-step function
= 1,
0,
● Ramp Function
unit-ramp function
= n,
0,
● Impulse
unit-pulse function
= 1,
0,
● Periodic Discrete-Time Signals
, for all integer
r: period ( fundamental period: smallest value of r for x[n] repeat)
example:
if periodic à
, or
if , , à fundamental period = 6
if , , à not a periodic signal
● Discrete-Time Rectangular Pulse
= 1,
0, all other n
● Digital Signals
: a discrete-time signal whose values belong to finite number of different values ;that is, each time instant for some i, where .
A sampled continuous-time signal is not necessarily a digital signal.
examples:
The sampled unit-ramp function r[n] is not a digital signal (since r[n] takes on an infinite range of values)
A binary signal is a digital signal whose values are equal to 1 or 0 only.
● Time-Shifted signals
指 shifted to right by steps(if )
指 shifted to left by steps(if )
example:
Right shift of p3[n]
Left shift of p3[n]
● Discrete-Time signals defined interval by interval
= ,
,
,
à ,