Window Functions

Window functions taper a finite signal block before an FFT so the block edges do not create a large discontinuity. That discontinuity causes spectral leakage: energy spreads into neighboring bins.

miniDSP provides five windows so you can compare the trade-off between main-lobe width (frequency resolution) and sidelobe level (leakage suppression).

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Hanning window

The Hanning (Hann) window is a common default:

\[w[n] = 0.5 \left(1 - \cos\!\left(\frac{2\pi n}{N-1}\right)\right), \quad n = 0, 1, \ldots, N-1 \]

It tapers smoothly to zero at both ends and gives good all-around performance for FFT analysis.

Reading the formula in C:

// n -> N (window length), i -> n (sample index), out[i] -> w[n]
if (n == 1) {
    out[0] = 1.0;
} else {
    double n_minus_1 = (double)(n - 1);
    for (unsigned i = 0; i < n; i++) {
        out[i] = 0.5 * (1.0 - cos(2.0 * M_PI * (double)i / n_minus_1));
    }
}

API:

void MD_Gen_Hann_Win(double *out, unsigned n);

Visuals — window taps and magnitude response:

The smooth taper to zero at both ends reduces leakage compared with a rectangular window.

Quick example:

    MD_Gen_Hann_Win(hann, N);

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Hamming window

The Hamming window keeps a similar shape to Hanning, but with non-zero endpoints and a lower first sidelobe:

\[w[n] = 0.54 - 0.46 \cos\!\left(\frac{2\pi n}{N-1}\right) \]

Reading the formula in C:

// n -> N (window length), i -> n (sample index), out[i] -> w[n]
if (n == 1) {
    out[0] = 1.0;
} else {
    double n_minus_1 = (double)(n - 1);
    for (unsigned i = 0; i < n; i++) {
        out[i] = 0.54 - 0.46 * cos(2.0 * M_PI * (double)i / n_minus_1);
    }
}

API:

void MD_Gen_Hamming_Win(double *out, unsigned n);

Visuals — window taps and magnitude response:

Compared with Hanning, the non-zero endpoints and coefficients shift the sidelobe pattern while keeping a similar main-lobe width.

Quick example:

    MD_Gen_Hamming_Win(hamming, N);

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Blackman window

The Blackman window strongly suppresses sidelobes by adding another cosine term:

\[w[n] = 0.42 - 0.5 \cos\!\left(\frac{2\pi n}{N-1}\right) + 0.08 \cos\!\left(\frac{4\pi n}{N-1}\right) \]

Compared with Hanning/Hamming, it has much lower sidelobes but a wider main lobe.

Reading the formula in C:

// n -> N (window length), i -> n (sample index),
// p -> 2*pi*n/(N-1), out[i] -> w[n]
if (n == 1) {
    out[0] = 1.0;
} else {
    double n_minus_1 = (double)(n - 1);
    for (unsigned i = 0; i < n; i++) {
        double p = 2.0 * M_PI * (double)i / n_minus_1;
        out[i] = 0.42 - 0.5 * cos(p) + 0.08 * cos(2.0 * p);
    }
}

API:

void MD_Gen_Blackman_Win(double *out, unsigned n);

Visuals — window taps and magnitude response:

You should see much lower sidelobes than Hanning/Hamming, with a wider main lobe in the response plot.

Quick example:

    MD_Gen_Blackman_Win(blackman, N);

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Rectangular window

The rectangular window is the no-taper baseline:

\[w[n] = 1 \]

It preserves the narrowest main lobe but has the highest sidelobes.

Reading the formula in C:

// i -> n (sample index), out[i] -> w[n]
for (unsigned i = 0; i < n; i++) {
    out[i] = 1.0;
}

API:

void MD_Gen_Rect_Win(double *out, unsigned n);

Visuals — window taps and magnitude response:

As the no-taper baseline, rectangular gives the narrowest main lobe and the highest sidelobes.

Quick example:

    MD_Gen_Rect_Win(rect, N);

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Kaiser window

The Kaiser window uses the zeroth-order modified Bessel function \(I_0\) to provide continuous control over the sidelobe/mainlobe tradeoff via a single parameter \(\beta\):

\[w[n] = \frac{I_0\!\left(\beta\,\sqrt{1 - \left(\frac{2n}{N-1}-1\right)^2}\right)} {I_0(\beta)}, \quad n = 0, 1, \ldots, N-1 \]

Higher \(\beta\) values produce lower sidelobes (better leakage suppression) at the cost of a wider main lobe:

• \(\beta \approx 5\) : ~45 dB stopband attenuation
• \(\beta \approx 10\) : ~100 dB stopband attenuation
• \(\beta \approx 14\) : ~120 dB stopband attenuation

Reading the formula in C:

// n -> N (window length), i -> n (sample index), out[i] -> w[n]
// beta -> shape parameter, inv_i0_beta -> 1/I₀(β)
if (n == 1) {
    out[0] = 1.0;
} else {
    double inv_i0_beta = 1.0 / MD_bessel_i0(beta);
    double n_minus_1 = (double)(n - 1);
    for (unsigned i = 0; i < n; i++) {
        double t = 2.0 * (double)i / n_minus_1 - 1.0;
        double arg = 1.0 - t * t;
        if (arg < 0.0) arg = 0.0;
        out[i] = MD_bessel_i0(beta * sqrt(arg)) * inv_i0_beta;
    }
}

API:

void MD_Gen_Kaiser_Win(double *out, unsigned n, double beta);

Visuals — window taps and magnitude response ( \(\beta = 10\)):

With \(\beta = 10\), the Kaiser window achieves ~100 dB stopband attenuation — much deeper suppression than Blackman, with a comparable main lobe width.

Quick example:

    MD_Gen_Kaiser_Win(kaiser, N, 10.0);

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Quick comparison

Window	Edge values	Sidelobes	Main lobe	Tunable?
Rectangular	1.0	Highest	Narrowest	No
Hanning	0.0	Low	Medium	No
Hamming	0.08	Lower first sidelobe	Medium	No
Blackman	0.0	Very low	Widest	No
Kaiser	> 0	Configurable via \(\beta\)	Configurable	Yes

If you are unsure where to start, Hanning is a good default. Use Blackman when leakage suppression matters more than peak sharpness. Use Kaiser when you need precise control over the sidelobe level (e.g., for FIR filter design or resampling). All response plots above use the same tap length and zero-padded FFT size, so sidelobe and main-lobe differences are directly comparable.

Further reading

• Window function – Wikipedia
• Hann function – Wikipedia
• Hamming window – Wikipedia
• Blackman window – Wikipedia
• Kaiser window – Wikipedia

API reference

• MD_Gen_Hann_Win() – generate a Hanning window
• MD_Gen_Hamming_Win() – generate a Hamming window
• MD_Gen_Blackman_Win() – generate a Blackman window
• MD_Gen_Rect_Win() – generate a rectangular window
• MD_Gen_Kaiser_Win() – generate a Kaiser window with configurable \(\beta\)