TIME-FREQUENCY RESOLUTION

The famous ``uncertainty principle'' of quantum mechanics resulted from observations that subatomic particles behave like waves with wave frequency proportional to particle momentum. The classical laws of mechanics enable prediction of the future of a mechanical system by extrapolation from presently known position and momentum. But because of the wave nature of matter with momentum proportional to frequency, such prediction requires simultaneous knowledge of both the location and the frequency of a wave. A sinusoidal wave has a perfectly clearly determined frequency, but it is spread over the infinitely long time axis. At the other extreme is a delta function, which is nicely compressed to a point on the time axis but contains a mixture of all frequencies. A mathematical analysis of the uncertainty principle is thus an analysis relating functions to their Fourier transforms.

Such an analysis begins by definitions of time duration and spectral bandwidth. The time duration of a damped exponential function is infinite if by duration you mean the span of nonzero function values. However, for nearly all practical purposes the time span is chosen as the time required for the amplitude to decay to e^-1 of its original value. For many functions the span is defined by the span between points on the time or frequency axis where the curve (or its envelope) drop to half of the maximum value. The main idea is that the time span $\Delta t$ or the frequency span $\Delta\omega$should be able to include most of the total energy but need not contain all of it. The precise definition of $\Delta t$ and $\Delta\omega$ is somewhat arbitrary and may be chosen to simplify analysis. The general statement is that for any function the time duration $\Delta t$ and the spectral bandwidth $\Delta\omega$ are related by  

$$\Delta\omega \; \Delta t \quad\geq\quad 2\pi$$ (1)

Although it is easy to verify (1) in many special cases, it is not very easy to deduce (1) as a general principle. This has, however, been done by D. Gabor. He chose to define $\Delta t$ and $\Delta\omega$ by second moments.

A similar and perhaps more basic concept than the product of time and frequency spreads is the relationship between spectral bandwidth and rise time of a system response function. The rise time $\Delta t$of a system response is also defined somewhat arbitrarily, often as the time span between the time of excitation and the time at which the system response is half its ultimate value. In principle, a broad frequency response can result from a rapid decay time as well as from a rapid rise time. Tightness in the inequality (1) may be associated with situations in which a certain rise time is quickly followed by an equal decay time. Slackness in the inequality (1) may be associated with increasing inequality between rise time and decay time. Slackness could also result from other combinations of rises and falls such as random combinations. Many systems respond very rapidly compared to the rate at which they subsequently decay. Focusing our attention on such systems, we can now seek to derive the inequality (1) applied to rise time and bandwidth. The first step is to choose a definition for rise time. The choice is determined not only for clarity and usefulness but also by the need to ensure tractability of the subsequent analysis. I have found a reasonable definition of rise time to be  

$$\Delta t \eq {\int^{\infty}_0 b(t)^2\, dt \over \int^{\infty}_0 {1 \over t}\, b(t)^2\, dt}$$ (2)

where b(t) is the response function under consideration. The numerator is just a normalizing factor. The denominator says we have defined $\Delta t$by the first negative moment. For example, if b(t) is a step function, then the denominator integral diverges, giving the desired $\Delta t = 0$ rise time. If b(t)² grows linearly from zero to t₀ and then vanishes, the rise time $\Delta t$ is t₀/2, again a reasonable definition.

Although the Z transform method is a great aid in studying situations where divergence (as 1/t) plays a key role, it does have the disadvantage that it destroys the formal identity between the time domain and the frequency domain. Presumably this disadvantage is not fundamental since we can always go to a limiting process in which the discretized time domain tends to a continuum. In order to utilize the analytic simplicity of the Z transform we now consider the dual to the rise-time problem. Instead of a time function whose square vanishes identically at negative time we now consider a spectrum $\overline{B} (1/Z) B(Z)$ which vanishes at negative frequencies. We measure how fast this spectrum can rise after $\omega = 0$.We will find this to be related to the time duration $\Delta t$ of the complex time function b_t. More precisely, we will now define the lowest significant frequency component $\omega = \Delta \omega$ in the spectrum analogously to (2) to be  

$$\Delta \omega \eq {\int^{\infty}_{-\infty} \overline{B} B\, ... ...^{\infty}_{-\infty} {1 \over \omega} \overline{B} B\, d\omega }$$ (3)

Without loss of generality we can assume that the spectrum has been normalized so that the numerator integral is unity. In other words, the zero lag of the auto-correlation of b_t is +1. Then  

$${1 \over \Delta \omega} \eq \int^{\infty}_{-\infty} {1 \over \omega} \overline{B} B\, d\omega$$ (4)

Now we recall the bilinear transform which gives us various Z transform expressions for $(-i\omega)^{-1}$.The one we ordinarily use is the integral $(\ldots 1, 1, 1, 0.5, 0, 0, 0 \ldots )$.The pole right on the unit circle at Z = 1 causes some nonuniqueness. Because $1/i\omega$ is an imaginary odd frequency function we will take the desired expansion to be the odd function of time given by  

$$(-i\omega)^{-1} \eq {(\cdots - Z^{-2} - Z^{-1} + 0 + Z + Z^2 + \cdots) \over 2}$$ (5)

Converting (4) to an integral on the unit circle in Z transform notation we have

$$\begin{eqnarray} {1 \over \Delta \omega} &= & \int^{+\infty}_{-\infty} -i {1 \ov... ...ots )\, \overline{B} \left( {1 \over Z} \right) \, B(Z)\, d\omega\end{eqnarray}$$ (6)

But since this integral selects the coefficient of Z⁰ of its argument we have  

$${1 \over \Delta \omega} \eq {-i \over 2} \left[ (r_{-1} - r_1) + (r_{-2} -r_2) + \cdots \right]$$ (7)

where r_t is the autocorrelation function of b_t. This may be further expressed as  

$${1\over\Delta\omega} \eq \sum^{\infty}_{t = 1} {\rm Im} (-r_t) \quad\leq\quad \sum^{\infty}_{t = 1} \vert r_t\vert$$ (8)

The sum in (8) is like an integral representing area under the |r_t| function. Imagine the |r_t| function replaced by a rectangle function of equal area. This would define a $\Delta t_{\rm auto}$ for the |r_t| function. Any autocorrelation function satisfies |r_t| < r₀ and we have normalized r₀ = 1. Thus, we extend the inequality (8) by  

$${1 \over \Delta \omega} \eq \sum^{\infty}_{t = 1} \vert r_t\vert < \Delta t_{\rm auto}$$ (9)

Finally, we must relate the duration of a time function $\Delta t$ to the duration of its autocorrelation $\Delta t_{\rm auto}$.Generally speaking, it is easy to find a long time function which has short autocorrelation. Just take an arbitrary short time function and convolve it by a long and tortuous all-pass filter. The new function is long, but its autocorrelation is short. If a time function has n nonzero points, then its autocorrelation has only 2n -1 nonzero points. It is obviously impossible to get a long autocorrelation function out of a short time function. It is not even fair to say that the autocorrelation must lie under some tapering function. To construct a time function with as long an autocorrelation as possible, the best thing to do is to concentrate the energy in two lumps, one at each end of the time function. Even from this extreme example, we see that it is not unreasonable to assert that  

$$\Delta t \quad\geq\quad \Delta t_{\rm auto}$$ (10)

inserting into (9) we have the uncertainty relation  

$$\Delta t \Delta\omega \quad\geq\quad 1$$ (11)

The more usual form of the uncertainty principle uses the frequency variable $f = 2\omega$ and a different definition of $\Delta t$, namely time duration rather than rise time. It is  

$$\Delta t \Delta f \quad\geq\quad 1 \qquad (\Delta t \mbox{ is duration})$$ (12)

The choice of a $2\pi$ scaling factor to convert rise time to duration is indicative of the approximate nature of the inequalities.

EXERCISES:

1. Consider B(Z) = [1 - (Z/Z₀)ⁿ]/(1 -Z/Z₀) in the limit Z₀ goes to the unit circle. Sketch the time function and its squared amplitude. Sketch the frequency function and its squared amplitude. Choose $\Delta f$ and $\Delta t$.
2. A time series made up of two frequencies may be written as

   $$b_t \eq A \cos \omega_1 t + B\sin \omega_1 t + C \cos \omega_2 t + D\sin \omega_2 t$$

   Given $\omega_1$, $\omega_2$, b₀, b₁, b₂, b₃ show how to calculate the amplitude and phase angles of the two sinusoidal components.
3. Consider the frequency function graphed below.
    

   E4-1-3 Figure 1 Exercise 4.1.3.

   

   Describe the time function in rough terms indicating the times corresponding to 1/f₁, 1/f₂, 1/f₃. Try to avoid algebraic calculation. Sketch an approximate result.

PROBLEM FOR RESEARCH

Can you find a method of defining $\Delta\omega$ and $\Delta t$ of one-sided wavelets in such a way that for minimum-phase wavelets only the uncertainty principle takes on the equality sign?