COSC 4452 3.0 DSP 3/13/2006

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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

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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)

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Fourier Analysis of Signals Using DFT

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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.

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Effect of Windowing: Sinusoid signals δ δδδ

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Effect of Windowing

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Solutions: a better window

Kaiser Windows

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Effect of Spectral Sampling

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Spectral sampling may mislead

64 point DFT

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Zero-Padding doesn’t improve frequency resolution

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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

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Fast Linear Convolution using DFT/FFT

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Circular Convolution is linear convolution with aliasing

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Circular Convolution is linear convolution with aliasing

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Circular Convolution is linear convolution with aliasing: wrap-on-itself

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Time-Dependent Fourier Transform• Non-stationary Signals

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Time-Dependent Fourier Transform(Short-time Fourier Transform)

),[ λnX 2-D function

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Effect of Windowing

• Choice of window length is a trade-off between frequency resolution and time resolution.

• For any fixed n, windowing � periodic convolution

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Sampling in Time and Frequency

• Sample short-time FT in frequency

• Then Sample it in time

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Block Convolution: overlap-adding

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Block Convolution: overlap-saving

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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)

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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?

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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

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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