appendix g

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APPENDIX G: CONVOLUTION 
Appendices 593 
Convolution and Fourier transforms 
Consider the time-dependent functions g(t) and h(t) and their respective frequency- 
dependent Fourier transforms G(f) and H(f). The convolution of the two original 
functions (written g �9 h) is defined as 
OC 
g ,h - / g(t)h(t- r)dt 
- - CX2, 
(G1) 
where g , h is a function of the time lag, T, and g , h -h �9 g. There is a one-to-one 
relationship between the function g �9 h and the product of the Fourier transforms of 
the two functions such that 
g , h +-, G(f) . H(f) (G2) 
Known as the convolution theorem, (G2) states that the Fourier transform of the 
convolution is the product of the Fourier transforms of the individual functions. In 
other words, convolution in one domain equates to multiplication in the other domain. 
We further note that the correlation ofg and h [corr(g, h); see Section 5.3] is written as 
O(3 
corr(g,h) - / g(t + r)h(t)dt 
- - OC 
(G3) 
which is also a function of the lag z. As with convolution, we can form the transform 
pair 
corr(g, h) ~ G(f). H(f)* (G4) 
called the'~correlation theorem, where H(f)* is the complex conjugate of H(f ) and 
H(f)* = H( - f ) since we are restricting discussion to the usual case in which g and h 
are real functions. As this relationship indictates, multiplying the Fourier transform 
of one function by the complex conjugate of the Fourier transform of the other 
function yields the Fourier transform of their correlation. The correlation of a 
function with itself is called its autocorrelation (Section 5.3). 
Convolution of discrete data 
The analysis of geophysical data commonly involves the convolution of specially 
designed "data windows" (convolution functions or filters) with time series records in 
order to smooth the spectral estimates obtained from these data and to improve the 
statistical reliability of spectral peaks. Good filters are those that minimize unwanted 
spectral leakage associated with the filter's side lobes in the frequency domain. 
Consider a filter h(t~) applied to a discrete data series g(O, where the tj and tk (j, k = 
0, ...) are discrete times in the data series. The filter will have non-zero values over a 
short segment of the data to which it is being applied and will be zero elsewhere, 
yielding a single value for the central time of the filter for that specific piece of the 
594 Data Analysis Methods in Physical Oceanography 
data. The filter h(tk) typically has a central peak and falls off to zero on either side of 
the maximum. 
The convolution theorem can be extended to discrete time series as follows. Assume 
that the time series, g(tj), has duration N and is completely determined by the N values 
g(to), ... , g( tN - 1). The convolution of this function with the window, h(tk), is a 
member of the discrete Fourier transform pair 
N/2 
g(tj_k)h(tk) ~ GnH~ (G5) 
k=-N/2+l 
where G,, (n = 0, ..., N - 1) is the discrete Fourier transform of the time series g(tj) O 
- 0, ..., N - 1), and Hn (n - O, .. . , N - 1) is the discrete Fourier transform of the 
function h(tk), (k - O, ..., N - 1). The values of h(tk) typically span a small fraction of 
the full data range k = -N /2 + 1, .. . , N/2. 
In Figure G1, the original time series, g(tj) (we have chosen normalized monthly 
values of the Southern Oscillation Index, SOI) is shown at the top and the convolution 
function, h(tk), used to filter the time series is presented in the middle panel. Here, we 
have used a simple five-year long Hamming window [see equation (5.6.78)]. The 
window (filter) is symmetrical, uses 61 monthly weights (with non-zero first and last 
weights), and begins with the first month of the time series. 
The bottom panel in Figure G1 shows the convolution of h(tk) with g(tj). As the 
filtered result clearly demonstrates, h(tk) acts as a smoothing function that flattens out 
the "bumpiness" of g(tj), reducing sharp year-to-year changes in the normalized SOI. 
This smoothing depends on the duration and the shape of the window, h(tk). A more 
sharply peaked h(tk) would produce less time series smoothing, leaving more of the 
large year-to-year variability. The function, h(tk), has exactly the same purpose as a 
moving average, except that the weights of the filter (the filter coefficients) are 
specially designed to reduce side lobe spectral leakage problems. For the moving 
average, all weights are of equal value. The convolved data (bottom panel of Figure 
G1) consists of variations longer than five years. Note the extended period of El Nifio 
events (negative SIO) in the 1980s and 1990s. 
Convolut ion as truncation of an infinite t ime series 
An observed time series, x(t), can be considered a subset of an unlimited duration time 
series g(t), obtained by convolving g(t) with a rectangular window h(t) of the form 
10<t<_T (G6) 
h( t ) - 0 otherwise 
x(t) = h(t)g(t) (G7) 
As illustrated in Figure G2, the series x(t) can be defined as 
x(t) =h(t )g(t ) (G7) 
It follows that the Fourier transform of x(t) is the convolution of the Fourier 
transforms of h(t) and g(t), namely 
Normalized Southern Oscillation (SOl) 
5 f [ T ~ T I X r 
Appendices 
[ T T 
595 
.~ 0 
O 
z 
-5 
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 
0.04 
Normalized Hamming Window, N = 61 (months) 
...... 
I ! I I I I I I I 
0.03 
0.02 
0.01 
1950 1 
Moving Filter (Convolution) 
I I 
 ooo oo, 
Convolution of SOl and Hamming Window 
5 i ~ / .... ~ ...... , I i 
o~ 
t - 
"10 
._N 0 
E 
O 
Z 
Extended El Nifio Period 
1950 1960 1965 I 1975 1980 I 5 I 0 1995 0 0 2005 
Year 
Figure GI. Convolution of the monthly time series of normalized Southern Oscillation Index (see U.S. 
government website ftp.necp.noaa.gov/pub/cpc/wd52dg/data/indicies/...) using a 61-month Hamming 
window filter). Negative (positive) values of the index are associated with El Nifzo (La Ni~a) events. 
The convolution emphasizes the low-frequency variability of the El Ni~o-La Ni~a phenomenon in the 
equatorial Pacific. 
CX3 
X(f) = / H(~)G(f - ~)d~ 
--O(3 
(G8) 
In this case, multiplication in the time equals convolution in the frequency domain, 
596 Data Analysis Methods in Physical Oceanography 
x(t) 
,.o t . . . . . 
~ ~v ,v "~ v W v ~W~~W" 
0 T 
Figure G2. Sampling a time series segment of duration T. The measurement is analogous to application of 
a rectangular window, h(t), of amplitude 1.0 and duration T to an extensive time series g(t). 
whereas in the previous case we examined convolution in the time domain (as with a 
running average) and multiplication in the frequency domain. These concepts are 
essential for the application of data windows in both the time and frequency domains. 
Deconvo lu t ion 
Deconvolution is the process of reversing (undoing) the smoothing that took place 
during application of the "data window", either in the time or frequency domains. It is 
assumed in this case that the response function is known and the process of 
deconvolution requires only a reverse of the process described above. Thus, the 
equation for deconvolution follows from that for convolution presented in equation 
(G1).