Methods of Experimental Physics Semester 1 2005 Lecture 6: Spectral Densities and All That 6.1 Introduction Lets get stuck in straight away... 6.2 Power and Energy in a Signal The energy of a signal can be written as: ∞ -∞ h(t) 2 dt (1) This integral is sometimes not defined (i.e. is infinite) for some of the signals we want to consider. For example, all random signals as well as most periodic signals will approach infinity if we use this definition. This is potentially irritating and not very useful so to overcome these issues we simplify matters by ignoring the total energy in a signal as in Eq. 1, and instead consider the average power in a given finite sample of a signal i.e. the energy delivered in a fixed amount of time (the power): Definition 6.1 P ∝ lim T →∞ 1 2T T -T h 2 (t) dt 6.3 Parseval’s Theorem There is a theorem, named after Parseval that relates the signal energy to the Fourier Transform: Definition 6.2 ∞ -∞ h 2 (t) dt = ∞ -∞ |H(f )| 2 df You will note that we have used H(f ) and integrated with respect to f in this expression above. If we had used ω as the argument then the result would have been ∞ -∞ h 2 (t) dt =2π ∞ -∞ |H(ω)| 2 df which is a bit annoying. This is another of the advantages of making use of f as the argument of H(f ) instead of ω. 6.4 Convolution A useful relation one can describe between two functions is called the convolution which is defined as: conv(g,h)(t) ≡ [gh](t) ≡ ∞ -∞ g(t )h(t - t )dt (2) I demonstrate the effect of convolution of one rectangular pulse function against another in Figure 1. The specific steps that one can view for implementing a convolution are given in Figure 2, while the commutativity of the convolution function is shown in Figure 3. 6-1
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Methods of Experimental Physics Semester 1 2005
Lecture 6: Spectral Densities and All That
6.1 Introduction
Lets get stuck in straight away...
6.2 Power and Energy in a Signal
The energy of a signal can be written as:
h(t)2dt (1)
This integral is sometimes not defined (i.e. is infinite) for some of the signals we want to consider.For example, all random signals as well as most periodic signals will approach infinity if we use thisdefinition. This is potentially irritating and not very useful so to overcome these issues we simplifymatters by ignoring the total energy in a signal as in Eq. 1, and instead consider the average power in agiven finite sample of a signal i.e. the energy delivered in a fixed amount of time (the power):
Definition 6.1
P limT
12T
TT
h2(t) dt
6.3 Parsevals Theorem
There is a theorem, named after Parseval that relates the signal energy to the Fourier Transform:
Definition 6.2
h2(t) dt =
|H(f)|2 df
You will note that we have used H(f) and integrated with respect to f in this expression above. If wehad used as the argument then the result would have been
h2(t) dt = 2pi
|H()|2 df
which is a bit annoying. This is another of the advantages of making use of f as the argument of H(f)instead of .
6.4 Convolution
A useful relation one can describe between two functions is called the convolution which is defined as:
conv(g, h)(t) [g ? h](t)
g(t)h(t t)dt (2)
I demonstrate the effect of convolution of one rectangular pulse function against another in Figure 1.The specific steps that one can view for implementing a convolution are given in Figure 2, while thecommutativity of the convolution function is shown in Figure 3.
6-1
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Lecture 6: Spectral Densities and All That 6-2
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