The continuous time case¶
The continuous time Fourier transform¶
Definition¶
We saw above that any discrete sequence can be expressed exactly as an infinite sum of complex exponentials. The same kind of result exist for continuous time signals. Any $x(t)$ can be expressed as the Fourier integral $$ x(t) = \TFI{X(f)}{f}{t}, $$ where $$ X(f) = \TF{x(t)}{t}{f}. $$
The Fourier transform exists if the three sufficient conditions of Dirichlet are verified:
\begin{enumerate} \item $x(t)$ possesses a finite number of discontinuities on any finite interval, \item $x(t)$ possesses a finite number of maxima and minima on any finite interval, \item $x(t)$ is absolutely integrable, that is $$ \int_{-\infty}^{+\infty} |x(t)| ~\dr{t} < +\infty. $$ Indeed, if $x(t)$ is absolutely integrable, then $$ \int_{-\infty}^{+\infty} |x(t)~e^{-j2\pi ft}| ~\dr{t} < \int_{-\infty}^{+\infty} |x(t)| ~\dr{t} < +\infty $$ (since $|x(t)~e^{j2\pi ft}| = |x(t)|~|e^{j2\pi ft}| < |x(t)|$). \end{enumerate}Example - The Fourier transform of a rectangular pulse¶
Let us look at this sinc function (you may play with several values of the width):
%matplotlib inline
def sinc(x):
if isinstance(x,(int,float)): x=[x]
x=np.array(x)
out=np.ones(np.shape(x))
I=np.where(x!=0)
out[I]=np.sin(x[I])/x[I]
return out
def dsinc(x,L): # This is the "discrete time" cardinal sinus
if isinstance(x,(int,float)): x=[x]
x=np.array(x)
out=np.ones(np.shape(x))
I=np.where(x!=0)
out[I]=np.sin(x[I])/(L*np.sin(x[I]/L))
return out
N=1000
f=np.linspace(-0.5,0.5,400)
plt.plot(f,sinc(pi*6*f))
plt.grid(b=True)
Playing with values using a slider.
N=1000
f=np.linspace(-0.5,0.5,400)
out = widgets.Output()
#----- Callbacks des widgets -------------
@out.capture(clear_output=True, wait=True)
def pltsinc(value):
#clear_output(wait=True)
T = s.value
plt.plot(f,sinc(pi*T*f))
plt.grid(b=True)
plt.show()
s=widgets.FloatSlider(min=0, max=20, step=0.1, value=8)
pltsinc('Width')
s.observe(pltsinc,'value')
display(widgets.VBox([s, out]))
#alternatively
#interact(pltsinc,value=fixed(1), T=[0.1,10,0.1])
\textbf{The integral of } $\sinc{\pi f T}$ -- Using the fact the Dirichlet integral is $$ \int_0^{+\infty} \sinc{x} \dr{x} = \frac{\pi}{2}, $$ the symmetry of $\sinc{}$, and a change of variable, we obtain that $$ \int_{-\infty} ^{+\infty} T \sinc{\pi f T} \dr{f} = 1. $$
It is now useful to look at the limit cases.
- First, let $T\to +\infty$, that is let the rectangular pulse tends to a constant value. Its Fourier transform, $T\sinc{\pi f T}$ tends to a mass on zero, since all the zero crossings occurs at zero. Furthermore, the amplitude is proportionnal to $T$ and then goes to infinity. Hence, the Fourier transform of a constant is a mass with infinite amplitude, located at 0. As we noted above, the integral of $T \sinc{\pi f T}$ equals to 1, which implies that the integral of this mass at zero is 1. This Fourier transform is not a function in the classical sense, but a distribution, see also the Encyclopedia of mathematics. In fact, it is the generalization of the Dirac $\delta$ function we had in discrete time.It is called \textem{Dirac distribution} (or function) and we end with the following pair
- Second, consider a rectangular pulse with amplitude $1/T$ and width $T$. When $T\to O$, this pulse tends to a Dirac distribution, a mass at zero, with infinite amplitude but also with a unit integral. By the Fourier transform of a rectangular pulse (\ref{eq:tf-rect}), we obtain that the Fourier transform of a Dirac function is a unit constant
%matplotlib tk
from matplotlib.widgets import Slider
fig, ax = plt.subplots()
fig.subplots_adjust(bottom=0.2, left=0.1)
slider_ax = plt.axes([0.1, 0.1, 0.8, 0.02])
slider = Slider(slider_ax, "L/T", 0, 100, valinit=8, color='#AAAAAA')
L=10
f=np.linspace(-0.5,0.5,400)
line, = ax.plot(f,dsinc(pi*L*f,L), lw=2,label="Discrete time sinc")
line2, = ax.plot(f,sinc(pi*L*f), lw=2,label="Standard sinc")
#line2 is in order to compare with the "true" sinc
ax.grid(b='on')
ax.legend()
def on_change(L):
line.set_ydata(dsinc(pi*L*f,L))
line2.set_ydata(sinc(pi*L*f))
slider.on_changed(on_change)
Table of Fourier transform properties¶
Symmetries of the Fourier transform.¶
(table adapted from cv.nrao.edu)
Dirac impulse, representation formula and convolution¶
Dirac impulse¶
Recall that the Dirac impulse $\delta(t)$ satisfies $$ \delta(t) = \left\{ \begin{array}{ll} 0 & \text{ if $t \not = 0$}, \\ +\infty & \text{ for $t = 0$}, \end{array} \right. $$ and is such that $$ \int_{-\infty}^{+\infty} \delta(t) \dr{t} = 1. $$ \index{Dirac impulse|Definition|fin}
Representation formula¶
The Dirac impulse plays the role of an indicator function. In particular, we have $$ x(t) \delta(t-t_0)=x(t_0) \delta(t-t_0). $$ Consequently, $$ \int_{-\infty}^{+\infty} x(t) \delta(t-t_0) \dr{t} = x(t_0) \int_{-\infty}^{+\infty} \delta(t-t_0) \dr{t}=x(t_0). $$ Therefore, we always have
$$ \begin{cases} x(t) = { \int_{-\infty}^{+\infty} x(\tau) \delta(t-\tau) \dr{\tau} }\\ \text{with } {x(\tau) = \int_{-\infty}^{+\infty} x(t) \delta(t-\tau) \dr{t}}. \end{cases} $$This is nothing but the continuous-time version of the \textem{representation formula}.
The set of distributions $\{\delta_\tau(t) : \delta(t-\tau)\}$, forms an orthonormal basis and $x(\tau)$ can be viewed as a coordinate of $x(t)$ on this basis. Indeed, the scalar product between $x(t)$ and $\delta_\tau(t)$ is nothing but $$ x(\tau) = < x(t),\delta_\tau(t) >= \int_{-\infty}^{+\infty} x(t) \delta(t-\tau) \dr{t}, $$ and $x(t)$ is then given as the sum of the basis functions, weighted by the associated coordinates: $$ x(t) = \int_{-\infty}^{+\infty} x(\tau) \delta(t-\tau) \dr{\tau}. $$
Following the same approach as in the discrete case, we define the \textem{impulse response} $h(t)$ as the output of a linear invariant system to a Dirac impulse. By linearity, the output of the system to any input $x(t)$, expressed using the representation formula, is $$ \fbox{$\displaystyle{y(t) = \int_{-\infty}^{+\infty} x(\tau)h(t-\tau) \dr{\tau} = [x*h](t).}$} $$ \index{Convolution!Definition|fin} This is the time-continuous \textem{convolution} between $x$ and $h$, denoted $[x*h](t)$. It enables to express the output of the filter using only the input and the impulse response. This shows the importance of the impulse response as a description of the system. The other notions we studied in the discrete case, namely transfer function, Plancherel and Parseval theorems, etc, extends straightforwardly to the continuous case.