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Introduction
Chapter 1
Signals
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A signal is a function of time, e.g.,
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notation:
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f is the force on some mass
vout is the output voltage of some circuit
p is the acoustic pressure at some point
f, vout, p or f(.), vout(.), p(.) refer to the whole signal or
function
f(t), vout(1.2), p(t + 2) refer to the value of the signals at
times t, 1.2, and t + 2, respectively
for times we usually use symbols like t, t , t1, . . .
Signal Example
Real Signals
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AM radio signal
FM radio signal
cable TV signal
audio signal
NTSC video signal
10BT Ethernet signal
telephone signal
System
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a system transforms input signals into output signals
a system is a function mapping input signals into
output signals
we concentrate on systems with one input and one
output signal, i.e., single-input, single-output (SISO)
systems
notation:
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y = S(u) means the system S acts on input signal u to
produce output signal y
Block System
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systems often denoted by block diagram
boxes denote systems; arrows show inputs &
outputs
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lines with arrows denote signals (not wires)
special symbols for some systems
System Example
Signals and Systems
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Modeling the physical world
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Physical system (e.g., LRC circuit) – using
mathematical equation
Input/output signal – using mathematical function
Signals and Systems
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Example: LRC
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LRC represented by a mathematical Equation
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ordinary diff. eqn.
No sampling (continuous time system)
V(i) is a mathematical function
Signals and Systems - Examples
Different systems can be MODELED using the same mathematical function
Signals and Systems - Examples
Human speech production system — anatomy and block diagram
Signals and System Categorizations
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Continuous time (analog)
Discrete time (digital)
Systems Described in Differential Equations
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Many systems are described by a linear constant coefficient
ordinary differential equation (LCCODE)
Second Order Continuous System
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Second-order RC circuit
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Closed loop system
Remember:
v1-y = iR2
v1=iR2+y
and
i(t) =C dv/dt
Find the
mathematical
relationship in
terms of input
& output
Substitute:
The 2nd order diff eqn can be solved using characteristic equation or auxiliary equation
Continuous System Example
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A digital player/recorder
Analog/Digital
Converter
Analog Input
Sampling Signal
Processor
Reconstructed
Digital Signal
Digital/Analog
Converter
Digital Output
Sample Matlab Code To Generate
Signal on the Soundcard!
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%%%%%%%
% The following program will send a 500 Hz sine wave to analog
% output channel 1 for one second.
%%%%%%%
%%Open the analog device and channels
AO = analogoutput('winsound',0);
chan = addchannel(AO,1);
%% Set the sample rate and how long we will send data for
%% 44,100 Hz, 1 seconds of data
duration = 1; %in seconds
frequency = 500 %in Hz
SampleRate = 44100;
set(AO,'SampleRate',SampleRate)
set(AO,'TriggerType','Manual')
NumSamples = SampleRate*duration;
%% Create a signal that we would like to send, 500 Hz sin wave
x = linspace(0,2*pi*frequency,NumSamples);
y = tan(sin(1*x))' - sin(tan(1*x))';
%y = sin(x)';
%data = y
data = awgn(y,10,'measured'); % wite noise
%% Put the data in the buffer, start the device, and trigger
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putdata(AO,data)
start(AO)
trigger(AO)
%% clean up, close down
waittilstop(AO,5)
delete(AO)
clear AO
%% clean up, close down
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%% Now let's plot the function for 5 cycles
x = 0:.1:2*pi*5;
data = tan(sin(x)) - sin(tan(x));
plot(x,data)
%% Now let's add random noise
%y = awgn(data,10,'measured'); % Add white Gaussian noise.
y = sin(x)';
plot(x,data,x,y) % Plot both signals.
legend('Original signal','Signal with AWGN');