Audio Signal Processing Time to Frequency Mapping

1
Audio Signal Processing-- Time to Frequency
Mapping

• Shyh-Kang Jeng
• Department of Electrical Engineering/
• Graduate Institute of Communication Engineering

2
Outline

• Introduction
• Discrete Fourier Transform
• Time-Frequency Analysis
• Time Domain Aliasing Cancellation
• Frequency vs. Time Resolution

3
Frequency Domain Coding

• Subdivide the input signal into a number of
  frequency components and quantize these
  components separately
• Subdivision into frequency components removes
  redundancy in the input signal
• Number of bits to encode each frequency component
  can be variable, so that encoding accuracy can be
  placed in frequencies where is most needed

4
Discrete Fourier Transform

• Discrete Fourier Transform
• Inverse Discrete Fourier Transform

5
Fourier Transform by DFT
0
0
6
Fourier Transform by DFT (cont.)

• Fourier Transform
• Inverse Fourier Transform

7
Window Function
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Hanning Window

• Formula
• Fourier Transform of

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Sine Window

• Formulae
• Fourier transform of

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Windows
Sine
amplitude
Hanning
11
Fourier Transform of a Sine Wave with Various
Windows
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Overlap-Add Scheme
M
Transform
FD samples
TD samples
Inverse Transform
FD samples
TD samples
M
N-M
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Reconstruction

• Window input signal with analysis window
• Apply transform to the windowed signal
• Apply the inverse transform
• Window with the synthesis window

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Window Constraints
N-1
N-1-M
0
M
NM-1
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Perfect Reconstruction

• Assume that the analysis window is the same as
  the synthesis window
• Assume that the window is symmetrical
• Assume no quantization
• A possible window

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Overlapping and Required System Rate

• Overlap N-M samples
• Slide the window by M samples
• Perform an N-point transform to obtain N
  frequency samples
• Transmit N frequency samples every M time samples
• If there is no overlap, we need only to transmit
  N frequency samples every N time samples
• Thus the required system rate is higher than that
  of the no-overlapping case, because MltN

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Time-Domain Aliasing Cancellation (TDAC)

• Critically sampled system
• Overall rate at the output of the analysis stage
  is equal to rate of the input signal
• DFT transform
• A small amount of overlapping increases the
  required data rate
• TDAC transform
• Provides a critically sampled system with 50
  overlap between adjacent windows
• The time domain alias is cancelled during the
  overlap and add stage

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Perfect Reconstruction TDAC Transform
N/2
Transform
N/2 FD samples
N TD samples
Inverse Transform
N/2 FD samples
N TD samples
N/2
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Oddly Stacked TDAC (OTDAC)

• Modified discrete cosine transform (MDCT)
• Inverse modified discrete cosine transform (IMDCT)

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Perfect Reconstruction TDAC Transform

• Symmetric analysis and synthesis windows
• Identical analysis and synthesis windows
• Sine window

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Fast Implementation of MDCT

• Pre-twiddle
• Compute FFT
• Post-twiddle

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Fast Implementation of IMDCT

• Pre-twiddle
• Compute IFFT
• Post-twiddle

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Time vs. Frequency Resolution

• TDAC filters operate at a fixed block length
• Static time/frequency resolution
• Steady state signals
• Localize quantization noise in spectral domain
  region where it is not audible
• High frequency resolution is needed
• Transient-like signals
• Prevent quantization noise to spread in time
  regions where it can be audible
• Sharp time resolution is needed

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Adaptive TDAC Filters

• Short-term stationary signals
• Harpsichord, oboe, etc.
• Rapid amplitude signal changes
• Castanets, glockenspiel, etc.
• Switch between long blocks (high frequency
  resolution) and short blocks (high time
  resolution)
• Typical window length
• Long blocks N
• Short blocks N/8

25
Steady-State vs. Transient Block Selection