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COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 1
3/13/2006 COSC4452 3.0 DSP 1
No. 9No. 9Digital Signal Processing Digital Signal Processing
Based on DFT/FFT Based on DFT/FFT
Prof. Hui Jiang
Department of Computer Science and Engineering
York University
COSC4452.3 Winter 2006Digital Signal Processing
3/13/2006 COSC4452 3.0 DSP 2
Topics• Fourier Analysis of Signals Using DFT
• Fast Linear Convolution Using FFT
• Short-time Fourier Transform
• Fourier Analysis of Non-stationary Signals– Speech processing
• Application II: speech enhancement (project II)
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 2
3/13/2006 COSC4452 3.0 DSP 3
Fourier Analysis of Signals Using DFT
3/13/2006 COSC4452 3.0 DSP 4
Fourier Analysis of Signals Using DFT
• Spectrum analysis using DFT’s
• How to infer Sc() and X() based on V[k]?– since Only V[k] are
computable.
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 3
3/13/2006 COSC4452 3.0 DSP 5
Effect of Windowing: Sinusoid signals δ δδδ
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Effect of Windowing
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 4
3/13/2006 COSC4452 3.0 DSP 7
Solutions: a better window
Kaiser Windows
3/13/2006 COSC4452 3.0 DSP 8
Effect of Spectral Sampling
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 5
3/13/2006 COSC4452 3.0 DSP 9extend by zero-padding (128-DFT)
Spectral sampling may mislead
64 point DFT
3/13/2006 COSC4452 3.0 DSP 10
Zero-Padding doesn’t improve frequency resolution
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 6
3/13/2006 COSC4452 3.0 DSP 11
How? increasing window size
L=32, N=1024 L=42, N=1024
L=54, N=1024 L=64, N=1024
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Fast Linear Convolution using DFT/FFT
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 7
3/13/2006 COSC4452 3.0 DSP 13
Fast Linear Convolution using DFT/FFT
3/13/2006 COSC4452 3.0 DSP 14
Circular Convolution is linear convolution with aliasing
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 8
3/13/2006 COSC4452 3.0 DSP 15
Circular Convolution is linear convolution with aliasing
3/13/2006 COSC4452 3.0 DSP 16
Circular Convolution is linear convolution with aliasing: wrap-on-itself
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 9
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Time-Dependent Fourier Transform• Non-stationary Signals
3/13/2006 COSC4452 3.0 DSP 18
Time-Dependent Fourier Transform(Short-time Fourier Transform)
),[ λnX 2-D function
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 10
3/13/2006 COSC4452 3.0 DSP 19
Effect of Windowing
• Choice of window length is a trade-off between frequency resolution and time resolution.
• For any fixed n, windowing � periodic convolution
3/13/2006 COSC4452 3.0 DSP 20
Sampling in Time and Frequency
• Sample short-time FT in frequency
• Then Sample it in time
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 11
3/13/2006 COSC4452 3.0 DSP 21
Block Convolution: overlap-adding
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Block Convolution: overlap-saving
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 12
3/13/2006 COSC4452 3.0 DSP 23
View Time-dependent FT as a Filter-Bank
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Speech Processing
• Speech signal is one of the most sophisticated signals in nature.
• Speech Processing:– Speech coding
– Speech enhancement
– Speech recognition
– Speech synthesis (text-to-speech)
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 13
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Application II: Speech Enhancement
• clean speech x[n]• noisy speech y[n]• Environmental model: y[n]=x[n]+ε[n]
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Spectrum Subtraction
• Assumptions– noise is stationary
– subtract speech magnitude or energy spectrum
• How to estimate noise spectrum?
COSC 4452 3.0 DSP 3/13/2006
Prepared by Prof. Hui Jiang 14
3/13/2006 COSC4452 3.0 DSP 27
Speech EnhancementDemo (I): White noise
• SNR 25dB– Example 1: Noisy � Cleaned
– Example 2: Noisy� Cleaned
• SNR 15dB– Example 1: Noisy� Cleaned
– Example 2: Noisy� Cleaned
• SNR 9dB– Example 1: Noisy� Cleaned
– Example 2: Noisy� Cleaned
3/13/2006 COSC4452 3.0 DSP 28
Speech EnhancementDemo (II): Speech Babble Noise
• SNR 25dB– Example 1: Noisy � Cleaned
– Example 2: Noisy� Cleaned
• SNR 14dB– Example 1: Noisy� Cleaned
– Example 2: Noisy� Cleaned