Digital Systems: Hardware Organization and Design
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Transcript Digital Systems: Hardware Organization and Design
Signal and Systems
Introduction to Signals and
Systems
Introduction to Signals and
Systems
Introduction to Signals and
Systems as related to Engineering
Modeling of physical signals by
mathematical functions
Modeling physical systems by
mathematical equations
Solving mathematical equations
when excited by the input
functions/signals.
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Modeling
Engineers model two distinct physical
phenomena:
1. Signals are modeled by mathematical
functions.
2. Physical systems are modeled by
mathematical equations.
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What are Signals?
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Signals
Signals, x(t), are typically real
functions of one independent variable
that typically represents time; t.
Time t can assume all real values:
-∞ < t < ∞,
Function x(t) is typically a real
function.
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Example of Signals: Speech
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Example of Signals EKG:
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Example of Signals: EEC
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Categories of Signals
Signals can be:
1. Continuous, or
2. Discrete:
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T – sampling rate
f – sampling
frequency – 1/T
– radial
sampling
frequency – 2f=
2/T
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Signal Processing
Signals are often
corrupted by noise.
s(t) = x(t)+n(t)
Deterministic
signal
Want to ‘filter’ the
measured signal s(t) to
remove undesired noise
effects n(t).
Need to retrieve x(t).
Corrupting,
stochastic
noise signal
Signal Processing
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What is a System?
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Modeling Examples
Human Speech Production is driven
by air (input signal) and produces
sound/speech (output signal)
Voltage (signal) of a RLC circuit
Music (signal) produced by a musical
instrument
Radio (system) converts radio
frequency (input signal) to sound
(output signal)
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Speech Production
Human vocal tract as a system:
Driven by air (as input signal)
Produces Sound/Speech (as output signal)
It is modeled by Vocal tract transfer
function:
Wave equations,
Sound propagation in a uniform acoustic tube
Representing the vocal tract with simple
acoustic tubes
Representing the vocal tract with multiple
uniform tubes
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Anatomical Structures for
Speech Production
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Uniform Tube Model
cos l x c
u x, t
U g e jt
cos l c
c sin l x c
p x, t j
U g e jt
A
cos l c
Volume velocity, denoted as u(x,t), is defined as the
rate of flow of air particles perpendicularly through a
specified area.
Pressure, denoted as p(x,t), and
u(0, t ) U g ()e
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RLC Circuit
L
R
i(t)
C
v(t)
t
di (t )
1
L
Ri (t ) i ( )d v(t )
dt
C
Voltage, v(t) input signal
Current, i(t) output signal
Inductance, L (parameter of the system)
Resistance, R (parameter of the system)
Capacitance, C (parameter of the system)
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Newton’s Second Law in Physics
2
d x(t )
f (t ) M
dt 2
The above equation is the model of a
physical system that relates an object’s
motion: x(t), object’s mass: M with a force
f(t) applied to it:
f(t), and x(t) are models of physical signals.
The equation is the model of the physical
system.
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What is a System?
A system can be a collection of
interconnected components:
Physical Devices and/or
Processors
We typically think of a system as
having terminals for access to the
system:
Inputs and
Outputs
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Example:
Single Input/Single Output (SISO) System
Electrical
Network
Vin
+
Vout
-
+
-
Multiple Input/Multiple Output (MIMO) System
x1 (t)
x2 (t)
System
…
…
xp (t)
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y1 (t)
y2 (t)
yp (t)
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Example:
Alternate Block Diagram Representation of a
Multiple Input/Multiple Output (MIMO) System
x(t)
System
x1 t
x t
2
xt
x p t p1
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y(t)
y1 t
y t
2
y t
yq t q1
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System Modeling
Physical
System
Mathematical
Model
Model
Analysis
Design
Procedure
Model
Simulation
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Model Types
1. Input-Output Description
Frequency-Domain Representations:
Transfer Function - Typically used on ideal
Linear-Time-Invariant Systems
Fourier Transform Representation
Time-Domain Representations
Differential/Difference Equations
Convolution Models
2. State-Space Description
Time-Domain Representation
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Model Types
1. Continuous Models
2. Discrete Models
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End
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