Transfer function of double exponential smoothing?

Double exponential smoothing a.k.a. Holt-Winters smoothing tracks level and trend of a time series in coupled IIRs:
$\qquad$ In: $Y_t$, t = 0 1 2 ...
$\qquad$ State: $L_t, T_t \quad$ -- level and trend
$\qquad$ Out: estimate $\hat{Y}_{t+1}$

$\qquad$ Parameters: a, b (a.k.a. $\alpha, \beta$)
$\qquad$ Step equations:
$\qquad$ $\qquad L_t = a Y_t + (1 - a) (L_{t-1} + T_{t-1}) \quad$ -- level
$\qquad$ $\qquad T_t = b (L_t - L_{t-1}) + (1 - b) T_{t-1} \quad$ -- trend
$\qquad$ $\qquad \hat{Y}_{t+1} = L_t + T_t$

How can I calculate the transfer function of this smoother, for given $a$ and $b$ ?

(Two possible approaches:

• manipulate the step equations into ratio-of-polynomials form


• brute force iterate with input $e^{2 \pi i f t}$: converges slowly for small $a, b$.

The first of these is beyond me, the second unsatisfactory.)

Answer

I would use the $\mathcal{Z}$-transform. Let me change the notation slightly by using $n$ as the discrete time index, lower case letters for time domain quantities, $x[n]$ as the observed input data sequence, and $y[n]$ as the output sequence, which is an estimate of $x[n+1]$. With this notation, the time domain equations are

$$\tag{1}\begin{align} l[n]&=ax[n]+(1-a)(l[n-1]+t[n-1])\\ t[n]&=b(l[n]-l[n-1])+(1-b)t[n-1]\\ y[n]&=l[n]+t[n] \end{align}$$

Taking the $\mathcal{Z}$-transform of these 3 equations gives

$$\begin{align}\tag{2} L(z)&=aX(z)+(1-a)z^{-1}(L(z)+T(z))\\ T(z)&=b(1-z^{-1})L(z)+(1-b)z^{-1}T(z)\\ Y(z)&=L(z)+T(z) \end{align}$$

From the second equation in (2) we can express $T(z)$ in terms of $L(z)$:

$$T(z)=\frac{b(1-z^{-1})}{1-(1-b)z^{-1}}L(z)\tag{3}$$

Plugging (3) into the first equation of (2), we can express $L(z)$ in terms of $X(z)$:

$$L(z)\left[1-(1-a)z^{-1}\left(1+\frac{b(1-z^{-1})}{1-(1-b)z^{-1}}\right)\right]=aX(z)$$

from which, after some algebra, you get

$$L(z)=\frac{a\left[1-(1-b)z^{-1}\right]}{1+[a(1+b)-2]z^{-1}+(1-a)z^{-2}}X(z)\tag{4}$$

Finally, plugging (3) into the last equation of (2) gives

$$Y(z)=\left[1+\frac{b(1-z^{-1})}{1-(1-b)z^{-1}}\right]L(z)=\frac{1+b-z^{-1}}{1-(1-b)z^{-1}}L(z)$$

and combining with (4) results in a relation between the output $Y(z)$ and the input $X(z)$:

$$Y(z)=\frac{a(1+b-z^{-1})}{1+[a(1+b)-2]z^{-1}+(1-a)z^{-2}}X(z)=H(z)X(z)\tag{5}$$

where $H(z)$ is the desired transfer function, which is of course a second order IIR filter.