COMPARING NOISE REMOVAL IN THE

WAVELET AND FOURIER DOMAINS

Dr. Robert Barsanti

SSST March 2011, Auburn University

Overview

• Introduction• Transform Domain filtering• Basis Selection• Simulations and Results• Summary

Introduction

(1) It is widely known that the DFT has it shortcomings.

(2) We look at using the DWT on these signals.

(3) We also use entropy to explain why one basis may be best.

(4) Simulations of the performance of the proposed algorithm are presented.

Noise Removal

• Separate the signal from the noise

TRANSFORMATION

Noisy Signal

Signal

Noise

FOURIER vs. WAVELETS

• Fourier Analysis

•  The DFT

• Wavelet Analysis

• The DWT

n a

bnnx

abaW *)(

1),(

)(log,...,1,02 2 NJa J

N

knj x(n)

n = kX )

2exp()(

Some Typical Wavelets

Signal in the Time, Fourier, & Wavelet Domain

Signal + Noise in the Time, Fourier, & Wavelet Domain

Threshold De-noising

DWT or DFT

Threshold Denoise

x(n) y(n) IDWT or IDFT

Use

Thres = Threshold Method

-hard-soft

Wavelet Based Filtering

0 0.5 1-5

0

5

10S1 Signal + Noise

0 0.5 1-10

-8

-6

-4

-2

0DWT of S1

0 0.5 1-4

-2

0

2

4

6S2 Denoised Signal

0 0.5 1-10

-8

-6

-4

-2

0DWT of S2

THREE STEP DENOISING

1. PERFORM DWT

2. THRESHOLD COEFFICIENTS

3. PERFORM INVERSE DWT

Basis Selection

i

ii ppxH )/1log()(

Best Basis will concentrate signal energy into the fewest coefficients.

Use Signal Entropy H(x) defined in [9]

Where pi is normalized energy of ith component

Entropy

The entropy H(x) is bounded such that;

)log()(0 NxH

H(x) = 0 only if all the signal energy is concentrated in one coefficient.

H(x) = log(N), only if pi = 1/N for all i.

The decomposition with the smaller entropy corresponds to the better basis for threshold filtering.

Simulation

2))()((1

n

nynxN

MSE

DWT or DFT

Threshold Denoise

x(n) y(n) IDWT or IDFT

Simulation

- 3 simulated signal waveforms using 2^10 points.

- Many trials using different instances of AWGN were conducted at signal to noise ratios ranging from -5 dB to 10 dB.

- A sufficient number of trials were conducted to produce a representative MSE curve. Simulations for the all the filters used the same noise scale.

Entropy Table

Domain Signal 1 Signal 2 Signal 3

Time 6.63 4.86 6.46

Fourier 0.693 4.88 3.50

Wavelet 2.14 3.74 2.52

Wavelets vs. Fourier

Filtering signal 1 at 10 dB using the DFT MSE vs. SNR for signal 1.

Wavelets vs. Fourier

Filtering signal 3 at 10 dB using the DFT MSE vs. SNR for signal 3.

Wavelets vs. Fourier

Filtering signal 3 at 10 dB using the DFT MSE vs. SNR for signal 3.

Summary

(1)Discussed noise removal on signals using DFT and DWT.

(2) Use of signal entropy as a measure of the best basis.

(3)Simulations compared performance on simple signals.