Let x(t) be a finite energy signal with Fourier transform X(jω). Let E denote the energy of the signal. Thus E <. We have.

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1 Notes on the Uncertainty principle Let x(t) be a finite energy signal with Fourier transform X(jω). Let E denote the energy of the signal. Thus E <. We have E = x(t) dt = X(jω) dω, π where the equality of the two integrals comes from the Parseval identity. We define the time center of the signal as t c = E t x(t) dt. You can think of this as the center of gravity of the energy distribution in time. We define the frequency center of the signal as ω c = ω X(jω) dω. πe You can think of this as the center of gravity of the energy distribution in frequency. We define the temporal width of the signal as t = (t t c ) E x(t) dt. You can think of this as measuring the spread of the energy distribution in time, around the time center. We define the spectral width of the signal as ω = πe (ω ω c ) X(jω) dω.

2 You can think of this as measuring the spread of the energy distribution in frequency, around the frequency center. Note that temporal width and spectral width are often defined in different ways, depending on the context. What we have chosen is just one of many possible definitions. The uncertainty principle is the name given to the following theorem : Theorem : For every finite energy signal x(t) we have t ω. Some of the conceptual content of this theorem can be gathered by imagining what would happen if t were very small, i.e. if we are dealing with a signal that is highly localized in time. The theorem tells us that the spectral width would then have to be very large, i.e. we are unable to pin down the signal in frequency. Dually, if we have a finite energy signal that is highly localized in frequency it cannot possibly be also localized in time. This theorem is mathematically related to the Heisenberg uncertainty principle of quantum mechanics. The lower bound in the theorem is achieved by a Gaussian pulse which has Fourier transform x(t) = π e t, X(jω) = e Ω. We now sketch a proof of the uncertainty principle under the additional d assumption that x(t) is differentiable, x(t) has finite energy, and dt lim t x(t) t ± = 0.

3 Since we may assume without loss of generality that t c = 0 and ω c = 0, we will do so. d Since the Fourier transform of x(t) is jωx(jω), we may use Parseval s dt relation to write We therefore have ω X(jω) dω = d π dt x(t) dt. t ω = E E t x(t) dt d dt x(t) dt tx (t) d dt x(t)dt, () where the second step is from the Cauchy-Schwarz inequality, which says that for two finite signals a(t) and b(t) we have a(t) dt b(t) dt ( a(t)b(t) dt). ( This inequality is analogous to the familiar result that for complex vectors [a, a,... a d ] T and [b, b,... b d ] T, we have ) ( a i )( b i ) ( a i b i ). i= i= i= The integral in the right hand side of eqn. () can be written as tx (t) d dt x(t)dt = t d dt x(t) dt = [ t x(t) = E. ] x(t) dt 3

4 We thus have which proves the theorem. t ω E E 4 = 4, Interpretation of the uncertainty principle The engineering intuition afforded by the uncertainty principle is the following : if one has available a bandwidth of W in which to communicate, over a time period of T, i.e. one has available a time-bandwidth product of T W, the number of non-interfering signals one can hope to use is at most a constant times T W. This sentence should be interpreted loosely at this stage, since we have not defined what we mean by non-interfering. One calls this number the number of degrees of freedom available in the given time-bandwidth product. Precise mathematical formulations of this principle have been worked out. See [] for an overview and references, if interested. It turns out that the correct proportionality constant to use is not, as suggested by the theorem above, but. If we were to measure frequency in Hz. then the number of π degrees of freedom available in a time-bandwidth product of T W is T W. This is a widely quoted result. TDMA and FDMA It is interesting to examine the time division multiple access (TDMA) and frequency division multiple access (FDMA) strategies from the point of view of how efficiently they use a given time-bandwidth product. Since these strategies are duals of each other, let us just consider TDMA. Given an available time interval T we decompose it into intervals, each of some duration, say, and use signals that are time limited to duration to communicate in each time interval. Consider the pulse x(t) = Π( t ), where Π(t) = { if t < 0 otherwise This has energy E =, time center t c = 0, and temporal width, according 4

5 to our definition, of t =. The Fourier transform of this signal is X(jω) = sinc ω π. This has frequency center ω c = 0, but a simple calculation shows that its spectral width, according to our definition, is ω = ω sinc ω π π dω =. Thus, TDMA is a rather poor user of the available time-bandwidth product, according to our definitions. It is more customary to evaluate the temporal width of x(t) as (our notion of width is one-sided), and the spectral width of X(jω) as π (since a signal that is temporally limited to can be recovered from its Fourier transform sampled at frequencies that are multiples of π, and since sinc(n) = 0 for n 0, we can place these samples as coefficients of the sinc functions centered at multiples of π, without interference ). This then suggests the rule of thumb that TDMA (and dually, FDMA) can fit in about T W degrees of π freedom in an available time-bandwidth product of T W. References [] David Slepian, Some comments on Fourier analysis, uncertainty and modelling, SIAM Review, Vol. 5, No. 3, pp You may see this as written as T W in some books, but this is when W is measured in Hz. We measure W in radians/sec. The uncertainty principle with W measured in Hz. would be t f 4π, which is the form in which you will see it in []. 5