Quadratic Interpolation of Spectral Peaks

In quadratic interpolation of sinusoidal spectrum-analysis peaks, we replace the main lobe of our window transform by a quadratic polynomial, or ``parabola''. This is valid for any practical window transform in a sufficiently small neighborhood about the peak, because the higher order terms in a Taylor series expansion about the peak converge to zero as the peak is approached.

Note that, as mentioned in §D.1, the Gaussian window transform magnitude is precisely a parabola on a dB scale. As a result, quadratic spectral peak interpolation is exact under the Gaussian window. Of course, we must somehow remove the infinitely long tails of the Gaussian window in practice, but this does not cause much deviation from a parabola, as shown in Fig.3.36.

Referring to Fig.5.15, the general formula for a parabola may be written as

$$y(x) \mathrel{\stackrel{\Delta}{=}}a(x-p)^2+b$$ (6.29)

The center point $p$ gives us our interpolated peak location (in bins), while the amplitude $b$ equals the peak amplitude (typically in dB). The curvature $2a$ depends on the window used and contains no information about the sinusoid. (It may, however, indicate that the peak being interpolated is not a pure sinusoid.)

At the three samples nearest the peak, we have

$$\begin{eqnarray*} y(-1) &=& \alpha \\ y(0) &=& \beta \\ y(1) &=& \gamma \end{eqnarray*}$$

where we arbitrarily renumbered the bins about the peak $-1$ , 0, and 1. Writing the three samples in terms of the interpolating parabola gives

$$\begin{eqnarray*} \alpha &=& ap^2 + 2ap + a + b \\ \beta &=& ap^2 + b \\ \gamma &=& ap^2 - 2ap + a + b \end{eqnarray*}$$

which implies

$$\begin{eqnarray*} \alpha- \gamma &=& 4ap \\ \Rightarrow\quad p &=& \frac{\alpha-\gamma}{4a} \\ \Rightarrow\quad \alpha &=& ap^2 + \left(\frac{\alpha-\gamma}{2}\right) +a+(\beta-ap^2) \\ \Rightarrow\quad a &=& \frac{1}{2}(\alpha - 2\beta + \gamma) \\ \end{eqnarray*}$$

Hence, the interpolated peak location is given in bins^6.9 (spectral samples) by

$$\zbox {p=\frac{1}{2}\frac{\alpha-\gamma}{\alpha-2\beta+\gamma}} \in [-1/2,1/2].$$ (6.30)

If $k^\ast$ denotes the bin number of the largest spectral sample at the peak, then $k^\ast+p$ is the interpolated peak location in bins. The final interpolated frequency estimate is then $(k^\ast+p)f_s/N$ Hz, where $f_s$ denotes the sampling rate and $N$ is the FFT size.

Using the interpolated peak location, the peak magnitude estimate is

$$\zbox {y(p) = \beta - \frac{1}{4}(\alpha-\gamma)p.}$$ (6.31)