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] = *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 Lobe Roll-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 -2πxfi dt r(x) is zero outside ±T/2: R(f) = ∫ -T/2, T/2 1. e -2πfti dt 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) = (e πfTi - 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 HanningHamming 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] = cos(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 ] = a 0 – a 1 cos(2πn/(N-1)) – a 2 cos(4πn/(N-1)) where: a 0 = (1-α)/2 ; a 1 = ½ ; a 2 = α /2; α =0.16 Blackman-Harris: w[n] = a 0 –a 1 cos(2πn/(N-1))–a 2 cos(4πn/(N-1)) - a 3 cos(6πn/(N-1)) where: a 0 = ; a 1 = ; a 2 = ; a 3 = 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: Y n = 1/M ∑ i=0, M-1 x n-I Slower than the IIR version IIR version: y n = y n-1 + (x n – x n-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: Y 4 = ¼( ) = 3 ½ – Next value using the IIR version: y 4 = 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 definition: – The middle value of an ordered list – If there is no middle value, average the two middle values Median filter: Y n = median m=0,M-1 {x n-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: a n * M n + b n * M n ≠ (an + bn) * M n  Succeeds: α (a n )*M n = (α a n )*M n  Succeeds: A n * M n = b n, then A n+k * M n+k = b n+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.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 signal 4.Add the result of the recursive application back Single Smoothing Double Smoothing

Compare Median to Moving Average Filter