Heisenberg Uncertainty Principle

A time signal $x(t)$ contains the complete information in time domain, i.e., the amplitude of the signal at any given moment $t$. However, no information is explicitly available in $x(t)$ in terms of its frequency contents. On the other hand, the spectrum $X(f)={\cal F}[x(t)]$ of the signal obtained by the Fourier transform (or any other orthogonal transform such as discrete cosine transform) is extracted from the entire time duration of the signal, it contains complete information in frequency domain in terms of the magnitudes and phases of the frequency component at any given frequency $f$, but there is no information explicitly available in the spectrum regarding the temporal characteristics of the signal, such as when in time certain frequency contents appear. In this sense, neither $x(t)$ in time domain nor $X(f)$ in frequency domain provides complete description of the signal. In other words, we can have either temporal or spectral locality regarding the information contained in the signal, but never both.

• The short-time Fourier transform (STFT), also called windowed Fourier transform, can be used to address this dilemma. The signal $x(t)$ to be Fourier analyzed is first truncated by a time window function $w(t)$ which is zero outside a certain time interval $T$, such as a square or Gaussian window, before it is transformed to the frequency domain. Now any characteristics appearing in the spectrum will be known to be from within this particular time window. In time domain, the windowed signal is:
  
  $$x_w(t)=x(t) w(t)$$
  
  
  According to convolution theorem, this equation corresponds to the following in frequency domain:
  
  $$X_w(f)=X(f) * W(f)$$
  
  
  where $X_w(f)={\cal F}[x_w(t)]$ and $W(f)={\cal F}[ w(t) ]$ are the spectra of $x_w(t)$ and $w(t)$, respectively. Now we know that all frequency components present in the spectrum $X_w(f)$ exist inside the time window, and the narrower the time window, the better the temporal resolution. However, on the other hand, the spectrum $X_w(f)$ of the windowed signal is a blurred version of the true signal spectrum $X(f)$, due to the convolution with the spectrum $W(f)$ of the window. Moreover, we see that while the temporal resolution can be increased by a narrow window $w(t)$, the frequency resolution will be reduced due to the expanded spectrum $W(f)$. Similarly, a narrower $W(f)$ for better frequency resolution corresponds to a wider window causing poorer temporal resolution.

• Fourier series expansion. If we assume the windowed signal repeats itself outside the window, i.e., it becomes a periodic signal with period $T$. The spectrum of this periodic signal is discrete, weighted by the Fourier coefficients, with a gap $f_0=1/T$, the fundamental frequency, between every two consecutive frequency components, i.e.,
  
  $$T f_0 =1$$
  
  
  This relationship indicates that it is impossible to increase both the temporal resolution (reduced $T$) and the frequency resolution (reduced $f_0$). When one of the resolutions is improved, the other must suffer.

• The uncertainty principle describes the general phenomenon quantitatively, similar to the Heisenberg Uncertainty Principle in quantum physics which states that it is impossible to precisely measure both the position and momentum of a microscopic particle at the same time. The more precisely one of the quantities is measured, the less precisely the other is known.

  To show this, we borrow the concept of probability density function (PDF) from the probability theory. Any given time signal $x(t)$ can be treated as a PDF by normalization:
  

  $$p_x(t)=\frac{\vert x(t)\vert^2}{\vert\vert x(t)\vert\vert^2... ...rt x(t)\vert^2}{\int_{-\infty}^\infty \vert x(t)\vert^2 dt},$$
  
  
  where the denominator is the total energy of the signal $x(t)$ assumed to be finite; i.e., $x(t)$ is an energy signal. As $p_x(t)$ satisfies these conditions
  
  $$p_x(t)>0\;\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\int_{-\infty}^\infty p_x(t) dt=1,$$
  
  
  How the signal $x(t)$ spreads over time; i.e., the locality or the dispersion of $x(t)$, can be measured as the variance of this probability density $p_x(t)$:
  
  $$\sigma_t^2=\int_{-\infty}^\infty (t-\mu_t)^2 p_x(t) dt =... ...rt^2}\int_{-\infty}^\infty (t-\mu_t)^2 \vert x(t)\vert^2 dt,$$
  
  
  where $\mu_t$ is the mean of $p_x(t)$:
  
  $$\mu_t=\int_{-\infty}^\infty t p_x(t) dt =\frac{1}{\vert\v... ...t)\vert\vert^2}\int_{-\infty}^\infty t \vert x(t)\vert^2 dt.$$
  
  
  Similarly, in the frequency domain, the locality or dispersion of the spectrum of the signal can also be measured as
  
  $$\sigma_f^2=\frac{1}{\vert\vert X(f)\vert\vert^2}\int_{-\inf... ...rt^2}\int_{-\infty}^\infty (f-\mu_f)^2 \vert X(f)\vert^2 df.$$
  
  
  Here, we have used Parseval's identity $\vert\vert x(t)\vert\vert^2=\vert\vert X(f)\vert\vert^2$, and $\mu_f$ is defined as
  
  $$\mu_f=\frac{1}{\vert\vert X(f)\vert\vert^2}\int_{-\infty}^\infty f \vert X(f)\vert^2 df.$$
  
  
  The uncertainty principle:

  Let $X(f)={\cal F}[x(t)]$ be the Fourier spectrum of a given function $x(t)$ and $\sigma_t^2$ and $\sigma_f^2$ be defined as above. Then
  

  $$\sigma_t^2 \sigma_f^2 \ge \frac{1}{16\pi^2}.$$