In order to be able
to bridge this gap between discrete time
and continuous time, we need to learn a bit more about continuous time
signal processing. So far, we have concentrated on discrete time
signal processing. We know that subject very well. So our introduction to continuous
time signal processing will be carried out
mainly by way of analogy. We will take concept
that we know, and we will see how they extend to continuous
time signals. The fundamental ingredients in continuous time signal
processing are the following. Time now is represented by a real valued variable t whose physical
dimension is seconds. Signals are complex functions
of set real variable t. Finite energy is defined by square integrability
of the signals. Once we have a set of square
integrable functions, we can define a vector space
just like we did for square summable sequences
that we call L2_R. In this vector space, we can define
an inner product like so the inner product between
two elements of the space is just the integral from
minus infinity to plus infinity of the conjugate of the first function times
the second function. Now, you see the analogy is apparent with the inner product
in discrete time. We fundamentally
replace our summation with an integral and our index n by a variable
t and everything holds. The energy of a continuous time
signal is the square norm defined of course as
the self inner product of a signal with itself, given our previous definition
of the inner product. Processing in continuous
time is performed by means of analog linear
time-invariant filters. This filters are
completely characterized by their impulse response
which in this case is a complex valued function
of a real variable t. The output of a LTI continuous time filter is the convolution between the input and the impulse response
as shown here. Now, the convolution in continuous time has
the same formal structure as a convolution in
discrete time once we replace once again
our summation with an integral. So we have this formulation here, and when can express
the convolution like we did in discrete time as the inner product
of the conjugate of the impulse response
with the input. In the real world, the concept
of frequency is intuitive. It represents the number of times something repeats
per unit of time. So an oscillation for instance would have
a certain number of cycles per second. Frequency is expressed
therefore in hertz which is one over seconds. Or alternatively, we could
use angular frequency if we use a 2Pi
normalization factor in front of the frequency. We will use the letter F to
express normal frequency, and uppercase Omega for
angular frequency when needed. The inverse of the frequency
is the period, which indicates how long it takes for a periodic
event to repeat, and therefore is
measured in seconds. In discrete time,
the maximum angular frequency is plus or minus Pi. Remember the argument is, if we have a point that
rotates along the unit circle, in discrete-time we
only get snapshots. So if the initial
position is here, and at the next instant
in time we're here. Because we only get snapshots, we cannot be sure whether
the point just traveled an angle of Pi or
an angle of 3Pi, or any odd multiple of Pi. But in continuous time, we
have no such limitation. We know the actual speed of the point as it moves
around the unit circle, the movement is defined by
a function of a real variable. So this speed can be
arbitrarily high. This has an important
consequence when we extend the concept of Fourier analysis to continuous time signals. First of all, the definition
of the Fourier transform of a continuous time
signal is conceptually identical to the
definition of the DTFT. That is, the Fourier transform of a continuous time
signal x of t is the inner product
between the signal, and a continuous time complex exponential at frequency
f. In other words, we're measuring
the similarity between our signal and
an uncountable family of continuous time
complex exponential at all possible
frequencies f. But now, because there is
no upper limit on the value of the frequency for
a continuous time complex exponential, the Fourier transform
is no longer periodic. It's a function of
variable f that goes from minus infinity to plus infinity with potentially
infinite support. Incidentally,
the Fourier transform can be inverted via
this formula here. Here you have
a perfect symmetry between the analysis and the synthesis
formula which as you remember was not the case with a DTFT where you
had the summation to go from the time domain to the frequency domain and
in integral to go back. Sometimes, the Fourier transform for continuous time signals is defined in terms of the angular frequency rather
than the standard frequency, in which case the analysis
formula take this form here, and the synthesis formula has an explicit normalization
factor in front. Remember, angular frequency is simply the standard frequency
multiplied by 2Pi. One reason why you
would want to use this type of notation
is that it mirrors, the relationship between
the DTFT and the Z-transform. Here we have that
the Fourier transform is a restriction to the
imaginary axis of the Laplace transform where X
of s would be the integral of x of t e to the minus st. Let's
show an example for instance here x of t is exponential function
of the form e to the minus at square. It's a Gaussian shape. If we take the Fourier transform, we have the following expression square root
of Pi over a, times e to the minus Pi square
over af square. We have a Gaussian shape
in the frequency domain. This is actually the only case in which the time-domain
expression for a function is the same as the frequency domain
expression for the same function. We can extend the Fourier
transform formalism to functions that are not
strictly finite energy, the same way we did
in discrete time. So for instance,
the Fourier transform of a sinusoid at frequency f0 will be a pair of
dirac Deltas at f0 and minus f0. Please note that
this dirac Deltas are not the 2Pi periodic stream of pulses that we have
seen discrete time, their individual pulses
at f0 and minus f0. Now that we have
developed the concept of frequency representations
for continuous time signals, we can go back to our LTI continuous
time systems and derive the equivalent of the convolution theorem
for continuous time. So if you have a filter H
with impulse response h of t, the output of the filter in the frequency domain
will be given by the product of the Fourier
transform of the input, times the Fourier transform
of the impulse response and this again is called the frequency response
of the filter.