Coherent gain of a window

The coherent gain of a discrete window function is its mean coefficient:It describes the amplitude scaling of a coherent complex tone under the chosen DFT normalization. Window coefficients and their normalization must be specified; a window's name alone does not fix every finite-length convention.^[ref]

For $x_n=A{e}^{i(2\pi k_{0}n/N+\varphi )}$, multiply by $w_n$ and evaluate the unnormalized discrete Fourier transform at the exact tone bin:Thus $|X_w[k_0]|/(N|G_c|)$ recovers $A$ when $G_c\ne 0$. This result is exact for this complex-tone model. A real cosine also has a negative-frequency component; the usual doubled one-sided amplitude requires that its contribution to the chosen positive-frequency bin be absent or separately accounted for.

A rectangular window has $G_c=1$. The periodic Hann window $w_n=(1-\cos (2\pi n/N))/2$ has $G_c=1/2$ for $N>1$. Its coherent bin therefore has half the unwindowed amplitude before correction. The symmetric endpoint convention $w_n=(1-\cos (2\pi n/(N-1)))/2$, for $N>2$, instead gives $G_c=(N-1)/(2N)$. At $N=64$ this is $0.4921875$, so blindly multiplying by two is not the exact correction for that convention.

Coherent gain does not correct every amplitude error. An off-bin tone samples the window's frequency response away from its maximum, and a DC-removing window with zero sum has no usable gain correction at all. Coherent sampling, leakage and window normalization are distinct conditions.

Noise uses a different scaling. For independent equal-variance samples, a weighted DFT bin's noise variance is proportional to ${\sum }_n{w}_n^2$, not $({\sum }_n{w}_n{)}^2$. This motivates equivalent noise bandwidth in bins,which is $3/2$ for a periodic Hann window with $N>2$. The manufacturer's table lists both coherent gain and noise bandwidth rather than treating one as a substitute for the other.^[ref]