bg physiological signal processing 02 signals and systems2019psp mk 7778

Nguyễn Công PhươngPHYSIOLOGICAL SIGNAL PROCESSING Signals and Systems... Characterization and Representation of Discrete – Time Systems... Signals and Systems– Extracting information fr

Nguyễn Công Phương

PHYSIOLOGICAL SIGNAL PROCESSING

Signals and Systems

I Introduction

II Introduction to Electrophysiology

III.Signals and Systems

IV Fourier Analysis

V Signal Sampling and Reconstruction

VI The z-Transform

Signals and Systems

1 Characterization and Representation of

Discrete – Time Signals

2 Characterization and Representation of

Discrete – Time Systems

Signals and Systems

– Extracting information from the signal,

– Extracting information about the relationships of two (or more) signals, – Producing an alternative representation of the signal.

• Why signals are processed ? [Bruce, 2001]

– To remove unwanted signal components that are corrupting the signal

Signals and Systems

Signals and Systems

1 Characterization and Representation of

Discrete – Time Signals

a) Types of Signals

b) Discrete – Time Signals

2 Characterization and Representation of

Discrete – Time Systems

Types of Signals (1)

• Signal : a physical quantity varying as a

function (of time, space, etc.) and carrying

Value of s(t) is defined for

every value of time t

Value of s(t) is defined

only at discrete time

Types of Signals (4)

Deterministic signal Random signal

The future value is

predictable

The future value is unpredictable

Signals and Systems

1 Characterization and Representation of

Discrete – Time Signals

a) Types of Signals

b) Discrete – Time Signals

i Representation of a Discrete – Time Signal

ii Basic Discrete – Time Signals

2 Characterization and Representation of

Discrete – Time Systems

Discrete – Time Signals

• T (sampling period): the

interval between two

successive samples, measured

in seconds (s)

• f s (sampling frequency, 1/T):

the number of samples per unit

of time, measured in hertz (Hz)

2 1

N N

, [ ]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Function Table

Graph Sequence

Basic Discrete – Time Signals (1)

, [ ]

,

n n

, [ ]

Basic Discrete – Time Signals (2)

, [ ]

Basic Discrete – Time Signals (3)

Basic Discrete – Time Signals (4)

Basic Discrete – Time Signals (5)

Basic Discrete – Time Signals (6)

Signals and Systems

1 Characterization and Representation of Discrete

– Time Signals

2 Characterization and Representation of

Discrete – Time Systems

a) Properties of Discrete – Time Systems

b) Impulse Response

c) Convolution

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference

Equations f) Properties of LTI Systems

Discrete – Time Systems

of Discrete – Time Systems (2)

• Definition : A system is called causal if the

present value of the output does not depend on future values of the input

• That is, y[n 0 ] is determined by the values of

x [n] for n ≤ n 0 , only

• Ex 1 : y [n] = x[n] + 2x[n – 1] + x[n – 2]

• Ex 2 : y [n] = x[n – 1] + x[n] + x[n + 1]

of Discrete – Time Systems (3)

• Definition : A system is called stable, in the

Bounded-Input Bounded-Output (BIBO)

sense, if every bounded input signal results in

a bounded output signal

• That is, |x[n]| ≤ M x < ∞  |y[n]| ≤ M y < ∞

• A signal x[n] is bounded if there exists a

positive finite constant M x such that |x[n]| ≤ M x for all n

• Ex 3 : y [n] = x[n] + 2x[n – 1] + x[n – 2]

• Ex 4 : y [n] = 2 n x [n]

of Discrete – Time Systems (4)

• Definition : A system is called linear if and

only if for every real or complex constant a 1 ,

a 2 and every input signal x 1 [n] and x 2 [n]:

H {a 1 x 1 [n] + a 2 x 2 [n]} = a 1 H {x 1 [n]} + a 2 H {x 2 [n]}

• A.k.a the principle of superposition

• Ex 5 : y [n] = 2x[n]

• Ex 6 : y [n] = x 2 [n]

of Discrete – Time Systems (5)

• Definition : A system is called time – invariant or

fixed if and only if:

y [n] = H{x[n]}  y[n – n 0 ] = H{x[n – n 0 ]}

for every input x[n] and every time shift n 0

• That is, a time shift in the input results in a

corresponding time shift in the output

Signals and Systems

1 Characterization and Representation of Discrete

– Time Signals

2 Characterization and Representation of

Discrete – Time Systems

a) Properties of Discrete – Time Systems

b) Impulse Response

c) Convolution

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference

Equations f) Properties of LTI Systems

Signals and Systems

1 Characterization and Representation of Discrete

– Time Signals

2 Characterization and Representation of

Discrete – Time Systems

a) Properties of Discrete – Time Systems

b) Impulse Response

c) Convolution

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference

Equations f) Properties of LTI Systems

Convolution (1)

• A great need for the evaluation of the

performance of an LTI (Linear Time –

Invariant) system.

• The response of a system can be used to

evaluate its performance.

• This can be obtained from impulse responses

h [n] by using the convolution.

Impulse response

n

δ [ ] h n [ ]

Convolution (6)

Ex 9

1 2 3 4 5 1 2 1 [ ] { }, [ ] { }.

× 4

− = 4

Convolution (7)

Ex 9

1 2 3 4 5 1 2 1 [ ] { }, [ ] { }.

×

2 = 4

Signals and Systems

1 Characterization and Representation of Discrete

– Time Signals

2 Characterization and Representation of

Discrete – Time Systems

a) Properties of Discrete – Time Systems

b) Impulse Response

c) Convolution

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference

Equations f) Properties of LTI Systems

Signals and Systems

1 Characterization and Representation of Discrete

– Time Signals

2 Characterization and Representation of

Discrete – Time Systems

a) Properties of Discrete – Time Systems

b) Impulse Response

c) Convolution

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference

Equations

f) Properties of LTI Systems

Linear Constant – Coefficient Difference Equations (LCCDE)

• a k : feedback coefficients.

• b k : feedforward coefficients.

• If a k & b k are fixed, then the system is time – invariant.

• If a k & b k depend on n, then time – varying.

• N is the order of the system.

Signals and Systems

1 Characterization and Representation of Discrete –

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference Equations

f) Properties of LTI Systems

i Properties of Convolution

ii Causality and Stability

iii Convolution of Periodic Sequences

iv Response to Simple Test Sequences

Causality and Stability

• A linear time – invariant system with impulse

response h[n] is causal if:

• A linear time – invariant system with impulse

response h[n] is stable, in the bounded – input

bounded – output sense, if and only if the impulse response is absolutely summable, that is, if:

Signals and Systems

1 Characterization and Representation of Discrete –

d) Block Diagrams & Signal – Flow Graphs

e) Linear Constant – Coefficient Difference Equations

f) Properties of LTI Systems

ii Causality and Stability

iii Convolution of Periodic Sequences

iv Response to Simple Test Sequences

h k

=−∞

=  [ ]

Response to Simple Test

Response to Simple Test