Chapter 1

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A signal is a function of time, e.g.,
 f is the force on some mass
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 vout is the output voltage of some circuit p is the acoustic pressure at some point notation:
 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, . . .

 AM radio signal
 FM radio signal
 cable TV signal
 audio signal
 NTSC video signal
 10BT Ethernet signal
 telephone signal

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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:
 y = S(u) means the system S acts on input signal u to produce output signal y

 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

 Modeling the physical world
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Physical system (e.g., LRC circuit) – using mathematical equation
Input/output signal – using mathematical function

 Example: LRC
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LRC represented by a mathematical Equation
 ordinary diff. eqn.
 No sampling (continuous time system)
V(i) is a mathematical function

Different systems can be MODELED using the same mathematical function

Human speech production system — anatomy and block diagram

 Continuous time (analog)
 Discrete time (digital)

Systems Described in Differential Equations
 Many systems are described by a linear constant coefficient ordinary differential equation (LCCODE)

 Second-order RC circuit
 Closed loop system Find the mathematical relationship in terms of input
& output
Remember: v
1
-y = i
R2
 v
1
=i
R2
+y and i(t) =C dv/dt
Substitute:
The 2 nd order diff eqn can be solved using characteristic equation or auxiliary equation

 A digital player/recorder
Analog/Digital
Converter
Processor
Digital/Analog
Converter
Analog Input
Sampling Signal
Reconstructed
Digital Signal
Digital Output

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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
%% 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');