I am trying to implement several filters in Matlab for Fourier domain filtering. They are the cosine, Shepp-Logan, and Hann/Hamming window filters. Replot with frequency in octaves from concert A = 440 Hz Wlinspace (0, 1, 181).' DFT power spectra using Hann window Use Matlab spectrogram function to make the same plot Replot with frequency in octaves from concert A 440 Hz Block-average S from Nav adjacent overlapping windows, then replot Figure 2: Zoom in on first second and two octaves and pick out notes These filters are defined as multiplying the ramp filter by the cosine function, sinc function, and Hann/Hamming windows respectively. Figure 4 shows the frequency domain behavior of the Hanning, Hamming, Blackman, and rectangular (no taper) windows. The first side lobe of the Hamming is lower (i.e. Hamming is better) than the first side lobe of the Hanning , but the distant side lobes of the Hanning are lower than the Hamming (thus the Hanning is better in that regard).

Most popular window functions are similar bell-shaped curves.In signal processing and statistics, a window function (also known as an apodization function or tapering function ) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. A popular window function, the Hann window. Figure 2: Zoom in on first second and two octaves and pick out notesLoad data and see what we have load handel %this music dataset is built into MatlabNy = length(y) % Number of samples in the segmentPlay the data as music p8 = audioplayer(y,Fs) DFT power spectra using Hann window Nw = 256 W = 0.5*(1-cos(2*pi*(j-1)/Nw))' % Hanning window of length Nw: % SP toolbox: w=hanning(Nw,'periodic')Varw = 3/8 % Mean-square value of taper function w (used to % renormalize power spectrum). % Segment the data into 'full' and 'half' windows, arrayed in columnsNwinf = floor(Ny/Nw) % Number of 'full' windows starting each Nw samplesNwinh = nwinf - 1 % Number of 'half' windows (half-way between fulls)Yw = zeros(Nw,nwin) % Define array in which to insert windowed dataYw(:,1:2:nwin) = reshape(y(1:nwinf*Nw),Nw,nwinf) Yw(:,2:2:(nwin-1)) = reshape(y((1+Nw/2):(nwinf-0.5)*Nw),Nw,nwinh) % Taper each column (i. The data in each window)S = (abs(ythat/Nw).^2)/varw %Windowed power spectrumS = S(1:Nw/2,:) % Only keep nonnegative harmonics for plottingSdB = 10*log10(S) % log of spectrum in decibel-like unitsMw = 0:(Nw/2 - 1) % Nonnegative windowed harmonicsFw = Fs*Mw/Nw % 'Frequencies'(inverse periods in Hz) of these harmonicsTw = (1:nwin)*0.5*Nw/Fs % Center time of each window % Plot the power spectra (log or decibel scale) vs. Window time% Add offsets and padding so pcolor centers each cell on (tw,fw) % and last row/col of field is included in plotsTitle( 'Windowed spectral power of Messiah sample')Use Matlab spectrogram function to make the same plot = spectrogram(y,hann(Nw, 'periodic'),Nw/2,Nw,Fs) % Convert 1-sided power spectral density P2 (in power per Hz) % into two sided periodogram (in power per FFT frequencySdB2 = 10*log10(S2) % Convert spectral power to decibel scaleImage(t2,fw2,SdB2, 'CDataMapping', 'scaled') %CDataMapping=scaled % uses the range of values of SdB2 to make the color scale.Axis xy % Puts low frequencies at the bottomReplot with frequency in octaves from concert A = 440 Hz % Plot log2(freq) = octaves, but have to exclude m=1 (zero freq)Pcolor(twpad,log2(fwpad(2:end)/440),SdBpad(2:end,:))Ylabel( 'octaves above concert A (440 Hz)')Block-average S from Nav adjacent overlapping windows, then replot Nav = 4 % Number of overlapping windows used for averaging spectra% Note use of reshape for computationally efficient block averagingTwav = mean(reshape(tw(1:nwininav),Nav,ntav)) % Average timesSav = reshape(S(:,(1:nwininav)),Nw/2,Nav,ntav) Twavpad = -0.5*dtwav SavdBpad = SavdB(,) Pcolor(twavpad,log2(fwpad(2:end)/440),SavdBpad(2:end,:))Title(,' int2str(Nav) '-spectrum avg'])Figure 2: Zoom in on first second and two octaves and pick out notes figure(2)% Make same block-avg windowed power spectrum plot as above.Pcolor(twav-0.5*dtwav,log2((fw(2:end)-0.5*dfw)/440),SavdB(2:end,:))Title(, ' int2str(Nav) '-spectrum avg'])% Zoom in on the first 1 second and the two octaves above concert A% Add dotted lines showing frequency intervals centered on particular % notes of the musical scale.Plot(, (floor(ymin*12)+0.5:ceil(ymax*12)-0.5)'*/12, 'w-')Note = Text(0.95*xmax+0.

But that method also changes the frequency content of the signal by an effect called spectral leakage. The duration of the segments is determined in each application by requirements like time and frequency resolution. Thus, tapering, not segmentation, is the main purpose of window functions.The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values.

2.6.3 Approximate confined Gaussian window 2.5.3 Nuttall window, continuous first derivative A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero. Rectangle, triangle, and other functions can also be used. There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice.In typical applications, the window functions used are non-negative, smooth, 'bell-shaped' curves.

The same amount of leakage occurs whether there are an integer (blue) or non-integer (red) number of cycles within the window (rows 1 and 2). Any window (including rectangular) affects the spectral estimate computed by this method.Figure 2: Windowing a sinusoid causes spectral leakage. In general, the transform is applied to the product of the waveform and a window function. Alternatively, one might be interested in their spectral content only during a certain time period.In either case, the Fourier transform (or a similar transform) can be applied on one or more finite intervals of the waveform. However, many other functions and waveforms do not have convenient closed-form transforms.

High-dynamic-range windows are most often justified in wideband applications, where the spectrum being analyzed is expected to contain many different components of various amplitudes.In between the extremes are moderate windows, such as Hamming and Hann. That is because the noise produces a stronger response with high-dynamic-range windows than with high-resolution windows.At the other extreme of the range of window types are windows with high dynamic range but low resolution and sensitivity. High-resolution, low-dynamic-range windows such as the rectangular window also have the property of high sensitivity, which is the ability to reveal relatively weak sinusoids in the presence of additive random noise. Conversely, windows that can distinguish components with similar frequencies and amplitudes are called high resolution.The rectangular window is an example of a window that is high resolution but low dynamic range, meaning it is good for distinguishing components of similar amplitude even when the frequencies are also close, but poor at distinguishing components of different amplitude even when the frequencies are far away. Windows that are effective against the first type of interference, namely where components have dissimilar frequencies and amplitudes, are called high dynamic range. But if the frequencies are too similar, leakage can render them unresolvable even when the sinusoids are of equal strength.

Everything else is leakage, exaggerated by the use of a logarithmic presentation. The actual frequency of the sinusoid is indicated as '13' on the horizontal axis. Figure 2, row 3 shows a DTFT for a rectangularly-windowed sinusoid. But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum. 90 Discrete-time signals When the input waveform is time-sampled, instead of continuous, the analysis is usually done by applying a window function and then a discrete Fourier transform (DFT). That trade-off occurs when the window function is chosen.

For a known frequency, such as a musical note or a sinusoidal test signal, matching the frequency to a DFT bin can be prearranged by choices of a sampling rate and a window length that results in an integer number of cycles within the window.Figure 3: This figure compares the processing losses of three window functions for sinusoidal inputs, with both minimum and maximum scalloping loss. In row 4, it misses the maximum value by ВЅ bin, and the resultant measurement error is referred to as scalloping loss (inspired by the shape of the peak). So the figure depicts a case where the actual frequency of the sinusoid coincides with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample.

The more the leakage, the greater the bandwidth. It can be thought of as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B. It is usually expressed as an equivalent bandwidth, B. But they also tend to be highly correlated with the total leakage, which is quantifiable.

The height of the noise floor is proportional to B. A graph of the power spectrum, averaged over time, typically reveals a flat noise floor, caused by this effect.

Hamming Window Vs Frequency Matlab