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.
