Narrow-band-pass filtering and amplitude modulation

−Rcosv+T sinv2 +ecosv 1 +ecosv

dt (2.4)

where n is earth’s average orbital angular velocity, v is the angle between perihelion and the earth, andais earth’s mean distance from the sun. The equation is meaningful over at least the last 5Ma as eccentricity,e, is always greater than zero. Note that only those components of the impulse in the orbital plane (RandT) act to change$. The instantaneous frequency associated with the precession parameter,d$/dt, is the sum of the instantaneous frequency associated with the longitude of perihelion, d$⁰/dt, plus the (nearly) constant 1/25.8KY precession of the equinoxes term. As stated earlier, the longitude of perihelion has a mean frequency of d$⁰/dt= 1/97KY giving an averaged climatic precession parameter frequency of 1/25.8+1/97=1/20.4KY.

Importantly, Eq 2.4 shows that the longitude of perihelion is more susceptible to perturbations when the eccentricity is small. That is, one expects the magnitude of d$⁰/dt to be largest when the eccentricity is smallest. Figure 2-3 shows the instan-taneous amplitude and frequency associated with precession parameter over the last 5 million years (Ma) calculated using a Hilbert transform [see e.g Bracewell, 2001] of the orbital solution ofBerger and Loutre [1991]. As expected from Eq. 2.4, there is a clear relationship between large excursions in instantaneous frequency and low values of the eccentricity. To quantify this relationship, Figure 2-3 also shows the abso-lute deviations in instantaneous frequency, |d$⁰/dt−d$⁰/dt|, plotted against inverse eccentricity, 1/e. The squared-cross-correlations between the absolute frequency de-viations and inverse eccentricity over the last 5 Ma is 0.56, indicating that variations in eccentricity account for the majority of the frequency variability in the climatic precession parameter.

2.2.2 Narrow-band-pass filtering and amplitude modulation

In the last section it was shown that variations in eccentricity cause changes in both the amplitude and frequency of the precession parameter. In this section it is shown that frequency modulation can be transformed into an amplitude modulation as a result of standard narrow-band-pass filtering of a signal. In a geophysical context,

−5000 −4000 −3000 −2000 −1000 0 0.01

0.02 0.03 0.04 0.05

amplitude

−4000 −3000 −2000 −1000

0 0.05 0.1

frequency (cycles/KY)

time (KY) 10² 10³

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

absolute frequency deviation

inverse eccentricity

Figure 2-3: Top left, The instantaneous amplitude of the precession parameter which, by definition, is the eccentricity. Bottom left, the instantaneous frequency of the precession parameter measured in cycles per KY and estimated using a Hilbert transform. The average frequency of the precession parameter over the last 5Ma is 1/20.4KY. Significant excursions from the mean are observed to occur during times of low eccentricity, as expected from Eq. 2.4. Right, To highlight the coupling between eccentricity and the precession frequency, the absolute deviations in instantaneous frequency, |d$/dt−1/20.4KY|, are plotted against inverse eccentricity, 1/e. Note the plot is logarithmically scaled. A strong positive cross-correlation of 0.75 exists between the variability in |d$/dt−1/20.4KY| and 1/e.

narrow-band-pass filtering is often utilized to isolate a narrow-band signal of interest from the broad-band continuum or other narrow-band signals. Therefor it is natural that narrow-band filtering is often used to isolate precession variability in paleoclimate records [e.g. Imbrie et al., 1984; Imbrie et al., 1993]. It is shown below that when the signal of interest has a frequency modulation associated with it, the resulting filtered signal will have an amplitude modulation related to its frequency modulation.

To see how narrow-band-pass filtering can generate amplitude modulation it is first useful to review some aspects of amplitude and frequency modulated signals.

Consider a pure cosine, cos(2πtf1), of carrier frequency f1 multiplied by another cosine of frequency f2. Then,

µ(t) = cos(2πtf1) cos(2πtf2) = cos(2πt(f1+f2)) + cos(2πt(f1−f2)) (2.5)

here the carrier frequency, f1, is split into two new frequencies,f1±f2, in the process known as amplitude modulation. A power spectrum of µ(t) would display peaks not atf1, but at (f1±f2), that is, with two-sidebands.

If instead a cosine is frequency modulated by another cosine we have, µ(t) = cos(2πtf1+ 2πδcos(2πtf2))

= cos(2πtf₁) cos(2πδcos(2πtf₂)) + sin(2πtf₁) sin(2πδcos(2πtf₂)). (2.6) Using a simple identity [Olver 1962, Eqs 9.1.44-45] Eq. (2.6) is,

µ(t) = cos(2πtf1)[Jo(2πδf2) + 2 [0,1,2...] with the relative amplitudes determined by the strength of the modulation term and the displacement from the carrier frequency.

The relevant point to be drawn from Eq 2.7 is that generating a frequency modu-lated signal, which is not amplitude modumodu-lated, requires contributions from frequen-cies extending out to infinity. Thus, any narrow-band-pass filtering of a frequency modulated signal will produce some amplitude modulation. The exact form of the amplitude modulation will depend on the frequency modulation and on the specifi-cations of the narrow-band-pass filter which is employed.

To be specific, a version of Eq 2.6 is used with µ(t) = cos

where the frequency modulation term, w⁰, is non-negative and has a 100KY period in rough analogy with the eccentricity variability. The frequency modulated signal, µ(t), is shown in Figure 2-4. Also shown is the periodogram of µ(t) which displays side-bands at 1/21±k/100 for k = {0,1,2...} as predicted by Eq. 2.7. Up to this pointµ(t) has no amplitude modulation, but now consider the effects of narrow-band-pass filtering. For clarity, a simple filtering technique is adopted whereby the Fourier transform ofµ(t) is taken, all the Fourier coefficients outside of a frequency band 1/25

to 1/17KY are set to zero, and then the inverse Fourier transform of the modified Fourier coefficients gives the filtered signal, ˜µ(t). ˜µ(t) is shown in Figure 2-4 along with its periodogram.

The narrow-band-pass filtering suppresses spectral energy outside of the band between 1/25 to 1/17KY, leaving only two dominant frequency components in ˜µ(t) at 1/23KY and 1/23+1/100=1/19KY of nearly equal magnitude. Thus, the filtered signal can be approximated as

˜

Eq. 2.5 shows that the sum of two cosines can be re-written as an amplitude modulated signal; in this case giving

˜

One further modification is now necessary to relate the amplitude modulation of ˜µ(t) to the frequency modulation,w⁰. The amplitude modulation term, cos(^2πt₂₀₀), becomes negative whereas the instantaneous amplitude is typically defined as a positive quan-tity. The absolute value of the amplitude modulation term can be written,

AM =

where the relationship cos(f)² = cos(2f) + 1 was used. Thus, both the amplitude modulation, AM, and frequency modulation, w⁰, terms are positive and periodic at 100KY. Figure 2-4 shows the excellent correspondence when both these terms are plotted against one another.