Now that we know what happens when we raw sample a continuous time complex exponential, we're ready to figure out the effect of raw sampling on an arbitrary signal, whether it's band limited or not. So the setup is that we will have a continuous time signal. For clarity we will use a subscript c to indicate explicitly continuous time. We raw sample the signal using the sampling period T_s and we obtain a discrete time sequence X of n, where each sample is equal to the continuous time function computed at multiples of T_s. The question is, given the original spectrum of the continuous time signal, what is the spectrum of the discrete time sequence obtained by raw-sampling? The key idea is to exploit the knowledge that we have gained by examining what happens when we sample complex exponentials. Let's pick a sampling period T_s and set F_s as 1 over T_s as per usual, and we pick a complex exponential whose frequency F0 is less than F_s over 2. So we know that when we raw sample this complex exponential, we will get a discrete time sequence where the digital frequency Omega is equal to 2 pi F0 T_s, which is less than pi. Now, if we pick another complex exponential at a frequency F0 plus F_s, remember F_s is the reciprocal of the sampling period, and we sample this with the same sample period as before, we get that the discrete time complex exponential, we have a frequency equal to 2 pi, then multiplies F0 plus F_s multiplied by T_s. So we can split this into two terms. The first is 2 pi of 0 T_s n, and the second is 2 pi F_s T_s n. Now, F_s and T_n are reciprocals, so this is equal to one. So the second part here it's just simply a multiple of two pi and so we can eliminate it from the phase of the complex exponential. So what we have found once again is that, if we sum F_s or any integer multiple of F_s to the frequency of a complex exponential in continuous time and we sample it, we will always obtain a complex exponential in discreet time at the frequency that is equal to F0 times T_s. Why is this intuition important? Because now if we have the sum of two distinct complex exponentials whose frequencies are F_s hertz apart and we sample the sum of complex exponentials with a period T_s, the result will be a complex exponential whose amplitude is the sum of the amplitudes of the two distinct components. So this two components there are distinct in continuous time, collapse into the same spectral component in discrete time. This intuition is important because we know that, thanks to the Fourier transform, we can always express a continuous time signal as the combination of an infinite number of complex exponentials with varying amplitudes. So when we sample a continuous time signal with a sampling frequency F_s, all the spectral components are F_s hertz apart will collapse into the same frequency location in the DTFT of the sampled sequence. So back to our task, we want to find out the spectrum of a raw-sampled sequence starting from the original spectrum of the continuous time signal. To derive mathematically the exact shape of the spectrum, we will start by computing the inverse Fourier transform of the original spectrum which is indicated by the notation X_c of F. We will compute this inverse Fourier transform at the location t equal to n T_s. This will give us the value of the continuous time function in n T_s, which is equal to the sample number n in the discrete time sequence. Then we will manipulate the quantity inside this integral until the integral looks like an inverse DTFT, which will give us the value of the discrete time sequence in n once again. So since this expression will be equal to this expression, we will be able to derive the form of the spectrum of the sampled sequence starting from the original spectrum. As we said before, frequencies there are F_s hertz apart will be aliased, there will collapse onto the same location. So it makes sense to split the integral in the inverse Fourier transform into non-overlapping intervals, we have the entire frequency axis, and we will compute the integral independently on sections that are F_s hertz wide. So this full integral over the entire frequency axis becomes the sum for k that goes from minus infinity to plus infinity of the same integral but computed over an interval that goes from k F_s minus F_s over 2, to k F_s plus F_s over 2. So say for instance k equal to 2, we will be into this integral. This is k F_s, and we will be computing this portion, so by arranging k over the entire set of integers, we cover entire frequency axis. Graphically what we're trying to do is collapse all this frequency contributions onto the baseband interval between minus F_s over 2 to F_s over 2. So now the value of X_n is given by this sum of integrals over this smaller intervals. We now operate a change of variable. We go from f to f plus k F_s. This corresponds to, instead of integrating over different sections, we always integrate between minus F_s over 2 to F_s over 2 but we slide the function to integrate over this interval. Now, with this change of variable, we have that e to J 2 pi f minus k F_s T_s n becomes simply e to the J 2 pi f T_s n because F_s T_s is equal to 1. So with this change of variable, we move from the original formulation to this formulation where the integration limits do not depend on k, and now we can interchange summation and integral. We have an integral between minus F_s over 2 n F_s over 2 of the sum of multiple copies of the continuous time spectrum, each copy shifted by a multiple of F_s. This is a periodization of the original analog spectrum with period F_s. As a matter of fact, we can define this auxiliary function tilde X_c of f. We use the tilde notation to stress the periodicity of this function. This is given by the sum of all possible shifts of X_c by all multiples of F_s. As we will see later in more detail, suppose this is our original spectrum, the periodization with period F_s will place copies of the spectrum at all multiples of F_s. With this auxiliary function, the value of Xn becomes the integral over the minus F_s over 2 F_s over 2 interval of the periodized spectrum times e to the J 2 pi f T_s n. Now, we operate another change of variable, we set Omega equal to 2 pi f T_s, so that f is equal to Omega over 2 pi times F_s. So df becomes simply F_s over 2 pi the Omega. When we replace this into the integral, we get that the value of Xn is 1 over 2 pi, times the integral between minus pi and pi because of the change of variable, times F_s, times our periodized spectrum computed in Omega over 2 pi times F_s, times e to the J Omega n d Omega. But this is indeed an inverse DTFT. This function here that was periodic in F_s in the previous slide, is now periodic in two pi. So it is indeed a valid inverse DTFT because the argument is a two pi periodic function of Omega. So if this is the inverse DTFT of this quantity, it means that this is the spectrum of the discrete time sequence obtained by raw sampling an arbitrary continuous time function. So once again the spectrum of the raw sampled sequence is equal to the periodized spectrum of the original continuous time signal scaled so that the minus F_s to F_s interval is mapped to the minus pi to pi interval explicitly. This is the sum of an infinite number of copies of the original spectrum shifted by all multiples of F_s and re-scaled to fit on the minus pi pi interval. This is a mouthful in formulas but it's very intuitive graphically. So let's have a look. Let's take a bandlimited signal for instance, in this case band-limited at some frequency F0, and we choose a sampling frequency so that F_s is larger than two times F0. So larger than twice the maximum positive frequency of the signal. The auxiliary signal that we computed before till the X_c of F, the periodized spectrum, is built by placing a copy of this spectrum here at all multiples of F_s. So you would have one copy here, one copy here and one copy here, and then by summing all these copies together. So what we get is this periodic repetition of the fundamental shape of the spectrum. Now, because we chose a sampling frequency that is larger than the maximum positive frequency. We don't have any overlap between this copies and so the periodization doesn't change the shape of the original signal. Then when we want to compute the spectrum of the raw sampled sequence, all we need to do is rescale this interval to the minus pi pi interval and we have the following shape. We can repeat the experiment where now the sampling frequency is exactly equal to twice the maximum positive frequency of the signal, this is called critical sampling. When we build the periodic repetition, we never really build the periodic repetitions like a way to visualize what's happening when we sample, the copies touched but do not overlap. So we still preserve the entire information contained in the spectrum of the original signal. When we map the minus F_s over 2 to F_s over 2 interval to the minus pi to pi interval in the digital domain, once again the shape of the spectrum is preserved. This is not the case if our sampling frequency is less than twice the maximum frequency. This is an example where F_s is slightly less than what should be necessary. So the periodized version of this spectrum will contain overlap when we put this copies at all multiples of F_s, the copies overlap and there sum will disrupt the shape of the original spectrum. Then the digital spectrum will be still the portion of this periodic spectrum between minus F_s over 2 and F_s over 2 map onto minus pi pi, but as you can see the shape is no longer like the original. We could not recover the original spectrum from the digital spectrum and this is because of aliasing. Aliasing also occurs if the original continuous time signal is not bandlimited. In this case no matter how high the sampling frequency, there will always be some overlap. If the signal tapers down, we can try and mitigate the effects of aliasing by choosing a larger sampling frequency but the repetitions will still overlap and there will be some distortion introduced by high-frequency aliasing. In this case you see that the floor of the signal gets raised because of the tails of this spectral shape and in continuous time, we will have a spectrum that will be very close to the original especially in the baseband area but not quite identical, we will now be able to recover the original spectrum from the digital spectrum.
