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.

Main Lobe

Side LobeRoll-off Rate

Spectral Leakage: Some of the spectral energy leaks into other frequency bins, which blur frequency distinctions.

Note: some windows act as a amplifier of total energy

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• 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 testsFails: 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 Smoothing1. Apply a median filter (ex:five point) and then a 2. Apply a linear filter (Ex: Hanning with values 0, ¼ , ½ , ¼, 0)3. Recursively apply steps 1 and 2 to the filtered signal4. Add the result of the recursive application back

Single Smoothing

Double Smoothing

Compare Median to Moving Average Filter