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Nguyễn Công PhươngPHYSIOLOGICAL SIGNAL PROCESSING Time – Frequency Representations... The z-Transform VII.Discrete Filters VIII.Random Signals IX... Spectral Analysis in Multiresolution.

Nguyễn Công Phương

PHYSIOLOGICAL SIGNAL PROCESSING

Time – Frequency Representations

I Introduction

II Introduction to Electrophysiology

III Signals and Systems

IV Fourier Analysis

V Signal Sampling and Reconstruction

VI The z-Transform

VII.Discrete Filters

VIII.Random Signals

IX Time-Frequency Representations

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

3 Spectral Analysis in Multiresolution

The Spectrogram

1

2 0

( / )

[ , ] [ ] [ ]

L

j k N m m

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

3 Spectral Analysis in Multiresolution

x n e N

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

a) Statistical Properties of the Periodogram

b) The Modified Periodogram

c) The Blackman – Tukey Method: Smoothing a

Single Periodogram d) The Bartlett – Welch Method: Averaging

Multiple Periodograms

3 Spectral Analysis in Multiresolution

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

a) Statistical Properties of the Periodogram

i Mean of the Periodogram

ii Variance & Covariance of the Periodogram

b) The Modified Periodogram

c) The Blackman – Tukey Method: Smoothing a Single

Periodogram d) The Bartlett – Welch Method: Averaging Multiple

Periodograms

3 Spectral Analysis in Multiresolution

Mean of the Periodogram (1)

N

j N

N

j N

Mean of the Periodogram (2)

1

1 1

N

j N

N

ω ω

Mean of the Periodogram (3)

1

1

1 2

2

N

π

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

a) Statistical Properties of the Periodogram

i Mean of the Periodogram

ii Variance & Covariance of the Periodogram

b) The Modified Periodogram

c) The Blackman – Tukey Method: Smoothing a Single

Periodogram d) The Bartlett – Welch Method: Averaging Multiple

Periodograms

3 Spectral Analysis in Multiresolution

Variance & Covariance

cov[ (2 I π k / N I ), (2 π k / N )] ≈ 0, k ≠ k

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

a) Statistical Properties of the Periodogram

b) The Modified Periodogram

c) The Blackman – Tukey Method: Smoothing a

Single Periodogram d) The Bartlett – Welch Method: Averaging Multiple

Periodograms

3 Spectral Analysis in Multiresolution

The Modified Periodogram (1)

21

The Modified Periodogram (2)

N c

−

=

The Modified Periodogram (3)

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

a) Statistical Properties of the Periodogram

b) The Modified Periodogram

c) The Blackman – Tukey Method: Smoothing a

The Blackman – Tukey Method

The Blackman – Tukey Method

The Blackman – Tukey Method

The Blackman – Tukey Method

The Blackman – Tukey Method

(5), Mean & Variance

The Blackman – Tukey Method

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

a) Statistical Properties of the Periodogram

b) The Modified Periodogram

c) The Blackman – Tukey Method: Smoothing a

The Bartlett – Welch Method (1)

1 0

The Bartlett – Welch Method (2)

1 0

The Bartlett – Welch Method (3)

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

3 Spectral Analysis in Multiresolution

a) The Short – Time Fourier Transform (STFT) b) The Continuous Wavelet Transform (CWT) c) The Discrete Wavelet Transform (DWT)

STFT (4)

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

3 Spectral Analysis in Multiresolution

a) The Short – Time Fourier Transform (STFT)

b) The Continuous Wavelet Transform (CWT)

c) The Discrete Wavelet Transform (DWT)

t t

s s

2 ( , ) ( , )

SC τ s = W τ s

t t

s s

e ω

ω ω

t t

s s

CWT (7)

Ex 3

CWT (8)

Ex 4

CWT (9)

Ex 5

CWT (10)

Ex 5

Time – Frequency Representations

1 The Spectrogram

2 The Periodogram

3 Spectral Analysis in Multiresolution

a) The Short – Time Fourier Transform (STFT)

b) The Continuous Wavelet Transform (CWT)

c) The Discrete Wavelet Transform (DWT)

DWT (1)

DWT (5)

Ex 2