Window functions compared

Five windows, one trade: watch the main lobe widen while the highest side lobe sinks.

Textbooks list dozens, and they sit on one trade

Windowing & spectral leakage (15.1) met three windows: rectangular, Hann and Blackman. Each one lowered the leakage of a tone, and each paid for it with a wider main lobe in the window’s own spectrum. The list in a textbook runs to dozens of windows. Every one of them is a point on that same trade: a wider main lobe buys lower side lobes.

Five numbers describe a window for measurement. Two are the pair from 15.1, the width of the main lobe and the height of the highest side lobe. Roll-off is how many dB the side lobes fall each time the distance from the tone doubles. Scalloping loss is how much a tone halfway between two bins reads low. Coherent gain and equivalent noise bandwidth say how the window scales a tone and noise. I will take them in that order, then put one knob on the whole trade, and end with which window to use for which measurement.

Every window on this page is the one SciPy makes by default, the periodic window. Hann is $w[n]=0.5-0.5\cos(2\pi n/N)$ for $n=0$ to $N-1$, so its main lobe is exactly 4 bins. NumPy’s np.hanning(N) divides by $N-1$ instead, which is the symmetric form made for filters. For spectra, use the periodic one.

Five windows, one trade

Two more windows join the three. Hamming is $w[n]=0.54-0.46\cos(2\pi n/N)$: Hann lifted off zero, so that the first side lobe nearly cancels. Flat-top (SciPy’s flattop) is a sum of five cosines,

with weights $a_0$ to $a_4$ of 0.21557895, $-0.41663158$, 0.277263158, $-0.083578947$ and 0.006947368. It is built for a main lobe with a flat top.

The picture at the top of this page steps through all five. When its clip has finished, press each window in turn and watch the bracket: it grows from 1 bin to 5, half of a main lobe that grows from 2 bins to 10, while the dotted level sinks from −13.3 dB to −93.0 dB. Hamming is the one to watch. Its first side lobe is lower than Hann’s, but its far side lobes are higher: a window trades near leakage for far leakage as well.

At a short length the numbers move a little. For $N=32$ the highest side lobes are $-13.2$, $-31.5$, $-41.8$, $-58.1$ and $-83.2$ dB for rectangular, Hann, Hamming, Blackman and flat-top, against $-13.3$, $-31.5$, $-42.7$, $-58.1$ and $-93.0$ at $N=1024$. The pictures and the table below use $N=1024$, which is the textbook large-$N$ case.

Between two bins, the peak bar sags

A tone does not sit on a bin unless it is lucky. Say a tone is $d$ bins from the nearest bin, with $d$ from 0 to 0.5. By Properties of the DTFT (12.3), the window’s spectrum $W$ stands in place of the tone’s arrow, so the tallest bar reads the main lobe at $d$, not at its top. The drop is the scalloping loss:

worst at $d=0.5$, halfway between two bins. A picket fence in front of a hill shows the same thing. The tallest picket is lower than the hilltop, unless the top is flat.

Watch the flat-top curve: it stays on 0 dB. Halfway between two bins, rectangular reads 3.92 dB low and Hann 1.42 dB.

The full curve is in numbers too. At $d=0.1$, 0.2, 0.25, 0.3 and 0.4, rectangular loses 0.14, 0.58, 0.91, 1.33 and 2.42 dB, and Hann loses 0.06, 0.22, 0.35, 0.51 and 0.91 dB. Flat-top stays within 0.01 dB all the way. At $d=0.5$ Hamming loses 1.75 dB and Blackman 1.10 dB.

Two scale factors

A tone of amplitude $A$ exactly on a bin reads $\lvert X[k]\rvert=A\sum_nw[n]/2$ (The DFT, 13.2, gave $AN/2$ for the plain cut, where every $w[n]$ is 1). The window’s average is its coherent gain:

For Hann it is 0.5. Divide a bin by $N\,\mathrm{CG}/2$ and you read $A$.

Noise is spread evenly over frequency (Oversampling and noise shaping, 11.3), so a bin collects noise in proportion to the area under $\lvert W\rvert^2$. In bins, that area is the equivalent noise bandwidth:

It is 1 bin for the rectangular window and 1.5 bins for Hann. The figure shows why.

Three more rows of the same measure. Read through Hann, Blackman and flat-top windows, the same white noise lands 1.76, 2.37 and 5.76 dB above its level through a rectangular window, when each tone is read with its own coherent gain. The bill for a wide, flat top is a bigger noise bin.

One knob: Kaiser’s β

The Kaiser window puts the whole trade on one dial. With the shape parameter $\beta$,

for $n=0$ to $N-1$, where $I_0(u)=\sum_{m\ge0}\big((u/2)^m/m!\big)^2$ is the modified Bessel function of order 0, a series that converges quickly. In SciPy it is get_window(('kaiser', beta), N). At $\beta=0$ the square root does not matter, because $I_0(0)=1$, and the window is the rectangular one.

Raising $\beta$ narrows the window in time, widens the main lobe and sinks the side lobes, smoothly. A camera’s aperture ring does the same with one ring: depth of field against light.

Watch the main lobe widen slowly while the side lobes fall fast. At $\beta=8.6$ the main lobe is close to Blackman’s width, with side lobes 5 dB lower.

For a side-lobe level you want, there is a rule. For a highest side lobe $a$ dB below the peak, Kaiser and Schafer give

for $a$ from 13.26 to 60, and

for $a$ above 60. The main lobe is then close to $2\sqrt{1+(\beta/\pi)^2}$ bins.

For 40 dB the rule gives $\beta=5.48$, and the window it makes reaches $-40.1$ dB with a main lobe of 4.02 bins. For 60 dB it gives $\beta=8.16$ and $-59.9$ dB. For 80 dB it gives $\beta=10.73$, $-79.9$ dB and a main lobe of 7.12 bins.

The window table

Everything so far goes into one table for $N=1024$ and periodic windows as SciPy makes them.

Fig. The five numbers for six windows (N = 1024, periodic windows as SciPy makes them). No row wins every column.

Choosing a window

Use the rule that fits the question.

• Tones exactly on bins, or a transient shorter than the record: rectangular. Nothing leaks when every tone is on a bin, and it has the narrowest main lobe and the lowest noise.
• A first look at an unknown signal: Hann.
• Weak tones near strong ones: Blackman, or Kaiser with $\beta$ from the side-lobe rule.
• Reading a tone’s amplitude: flat-top, with 0.01 dB of scalloping, at the cost of 3.77 bins of noise.
• Two close tones of similar size: the narrowest main lobe that the side lobes allow, so rectangular or Hann.

Worked example

1. Reading a 1 V tone halfway between bins. Through a rectangular window it reads 0.637 V, through Hann 0.849 V and through flat-top 0.999 V. Each is $10^{\text{loss}/20}$ for the loss at $d=0.5$.
2. Noise per bin. The ENBWs 1.5, 1.727 and 3.770 are 1.76, 2.37 and 5.76 dB.
3. Kaiser targets. $\beta$ of 0, 2, 4, 6, 8.6, 10 and 12 give main lobes of 2.00, 2.37, 3.24, 4.31, 5.83, 6.67 and 7.90 bins. Their highest side lobes are $-13.3$, $-18.4$, $-29.9$, $-43.8$, $-63.3$, $-74.1$ and $-89.9$ dB.
4. Short windows. At $N=32$ the highest side lobes are $-13.2$, $-31.5$, $-41.8$, $-58.1$ and $-83.2$ dB. At $N=1024$ they are $-13.3$, $-31.5$, $-42.7$, $-58.1$ and $-93.0$.

Where you’ll meet this

Zero-padding and resolution (15.3) separates the reading of the bars from the resolution of the window. Reading a spectrum: scaling and units (15.4) uses $\mathrm{CG}$ and $\mathrm{ENBW}$ to put numbers on a spectrum. Kaiser windows return in Window-method FIR design (19.1), where NumPy’s symmetric windows belong, and tapers for spectral estimation in The periodogram (25.1).

Reference card