Transcript Tricks - Southern Oregon University

Windowing
• Purpose: process pieces of a signal and
minimize impact to the frequency domain
• Using a window
– First Create the window: Use the window formula
to calculate an array of window values
– Next apply the window: multiply these values by
each time domain amplitude in a frame
• Window Length: The number of signal values
in a frame is the window length
Example: Hamming Window
• Create Window
double[] window = new double[windowSize];
double c = 2*Math.PI / (windowSize - 1);
for (int h=0; h<windowSize; h++)
window[h] = 0.54 - 0.46*Math.cos(c*h);
• Apply window
for(int i=0; i<window.length; i++)
frame[i]=frame[i]*window[i];
• Note: Time domain multiplication is frequency domain
convolution. Therefore, windows act as a frequency domain
filter.
Windowing Frequency Response
• Main Lobe: Narrow implies better frequency resolution. As the window
length grows, the main lobe width narrows (less initial spectral leakage).
• Side lobe: Higher for abrupt time domain window transitions from 1 to 0.
• Roll-off rate: Slower for abrupt window discontinuities.
Spectral Leakage:
Side Lobe
Roll-off Rate
Some of the spectral energy
leaks into other frequency
bins, which blur frequency
distinctions.
Note: some windows act
as a amplifier of total energy
Main Lobe
Rectangular Frequency Response
• Fourier transform of r(x): R(f) = ∫∞,∞r(x) e-2πxfidt
• r(x) is zero outside ±T/2: R(f) = ∫-T/2, T/2 1 . e-2πftidt
• Chain Rule: The integral of e-2πfti is e-2πfti /(-2πfi)
R(f) = e-2πfti /(-2πfti)|T/2,-T/2 = e-2πfT/2i/(-2πfi) - e-2πf(-T/2)i/(-2πfi)
• By Eulers formula: R(f) = (eπfTi - e-πfTi)/(2πfi) = sin(πfT)/(πfT)
Rectangular Window
• Main lobe width: 4π/M, Side lobe width: 2 π /M
• Minimal leakage, but directed to places that hurt analysis
• Poor roll off and high initial lobe
Convoluting Rectangular Window
with a particular frequency
Note: Windows with more points narrow the central lobe
Triangular (Bartlett Window)
Compare: Hamming to Hanning
Hanning
Hamming
Hanning: Faster roll of; Hamming: lower first lobe
Evaluation: Non-Rectangular
• Greater total leakage, but redistributed to places where
analysis is not affected.
• If little energy spills outside the main lobe, it is harder to
resolve frequencies that are near to each other.
• Fast roll-off implies wide initial lobe and higher side lobe;
worse at detecting weak sinusoids amidst noise.
• Moderate roll-off, implies narrower initial lobe and lower
side lobe.
• Tradeoff: Resolving comparable strength signals with
similar frequencies (moderate) and resolving disparate
strength signals with dissimilar frequencies (fast roll-off).
Windowing Formulae
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•
•
•
•
Hanning: w[n] = 0.5-0.5cos(2πn/(N-1))
Hamming: w[n] = 0.54 – 0.45 cos(2πn/(N-1))
Bartlett: w[n] = 2/(N-1).(N-1)/2 - |(n–(N-1)/2|)
Triangle: w[n] = 2/N . (N/2 - |(n–(N-1)/2|)
Blackman: w[n] = a0 – a1 cos(2πn/(N-1)) – a2 cos(4πn/(N-1))
where: a0 = (1-α)/2 ; a1 = ½ ; a2 = α /2; α =0.16
• Blackman-Harris:
w[n] = a0–a1cos(2πn/(N-1))–a2cos(4πn/(N-1)) - a3cos(6πn/(N-1))
where: a0=0.35875; a1=0.48829; a2=0.14128; a3=0.01168
Note: Small formula changes can cause large frequency
response differences
Moving Average Filters
• Both filters can be used to smooth a signal and reduce
noise
• It curves sharp angles in the time domain.
• It overreacts to outlier samples
• Slow roll-off destroys ability to separate frequency bands
• Horrible stop band attenuation
• Evaluation
– An exceptionally good smoothing filter
– An exceptionally bad low-pass filter
Moving Average Filter
• FIR version:
• Yn = 1/M ∑i=0, M-1 xn-I
• Slower than the IIR version
• IIR version:
• yn = yn-1 + (xn – xn-M)/M
• Propagates rounding errors
• Example: {1,2,3,4,5,4,3,2,1,2,3,4,5}; M = 4
– Starting filtered values: {¼ , ¾, 1 ½, 2 ½, … }
– Next value using the FIR version: Y4 = ¼(5+4+3+2) = 3 ½
– Next value using the IIR version: y4 = 2 ½ + (5 – 1)/4 = 3 ½
– Not appropriate for speech because:
– Blurs transitions between voiced/unvoiced sounds
– Negatively impacts the frequency domain
Median Filter
Median Filter
• Median definition:
– The middle value of an ordered list
– If there is no middle value, average the two middle values
• Median filter: Yn = medianm=0,M-1 {xn-m}
• Advantages
– Good edge preserving properties
– Preserves sharp discontinuities of significant length
– Eliminates outliers
• Applications
– The most effective algorithm to removes sudden impulse noise
– Remove outliers in estimates of a pitch contour
• Implementation: Requires maintaining a running sorted list
Characteristics: Median Filter
• Linearity tests
Fails: an * Mn + bn * Mn ≠ (an + bn) * Mn
Succeeds: α (an)*Mn = (α an)*Mn
Succeeds: An * Mn = bn, then An+k * Mn+k = bn+k
• Every output will match an input (if the filter
length is odd)
• Frequency response
• There is not a mathematic formula
• Must be determined experimentally
Double Smoothing
1.
2.
3.
4.
Apply a median filter (ex:five point) and then a
Apply a linear filter (Ex: Hanning with values 0, ¼ , ½ , ¼, 0)
Recursively apply steps 1 and 2 to the filtered signal
Add the result of the recursive application back
Single Smoothing
Double Smoothing
Compare Median to Moving
Average Filter