So upsampling increases the implicit sampling rate of a signal by creating new output samples for each input sample. The converse operation is called downsampling, and it reduces the implicit sampling rate by keeping only one output sample out of capital N input samples, or alternatively, by discarding n minus one input samples every N input samples. The fact that we're discarding samples should raise all sorts of alarms and indeed we have to be careful when we use downsampling because downsampling is affected by a discrete time version of the alleys in phenomenon as we will see shortly. Upsampling by contrast is a harmless operation because it only adds the samples which can be removed later on if necessary. Formerly, a downsampled sequence is obtained simply by retaining one sample out of capital N samples. The symbol for the downsampling operator is a circle with the downsampling factor and an arrow pointing downwards. Graphically, if you have a signal like so, and we downsample it by four, what we do is we retain only one sample out of four. So the orange samples here will be the survivors and the resulting signal will be the following. As you can see, there's a significant loss of information and we have to make sure that this loss of information is not irreversible. The spectral representation of a downsampled sequence is a little bit trickier to derive than in the case of upsampling. Like before, we start with the z transform of the downsampled sequence and the expression for the z transform is now the sum for k that goes from minus infinity to plus infinity of the surviving samples. So x of k capital N times z to the minus k. This expression is not really easy to take apart as is. So here comes the first trick. Consider an auxiliary z transform, A of z which is the sum for all values of k of x of k times capital N like before. But now, you multiply this by z to the minus k times capital N. If we knew the z transform A of z, then the z transform of the downsampled sequence would be just A computed in z to the power of one over capital N. Then when we bring this to the unit circle like before, we would have an expansion of the frequency axis by a factor of N before we had the contraction. So now the question is, how do we find the value of A of z? So here comes the second trick. A of z as we saw is the sum of x of kN times z_kN. So you see that all the indices here are multiplied by capital N. So we're hopping over multiples of capital N. We can rewrite this as the z transform of the original sequence x of k times an auxiliary sequence Xi of k that is used to kill all terms in the sum where k is not a multiple of capital N. So Xi of N is an indicator sequence that only selects samples whose index is a multiple of capital N. As an example, here's the graphical representation of Xi of n for capital N equal to four. You will have one sample that is equal to one followed by three zeros samples and so on so forth. Now here comes the third trick. We can remember the formula for the orthogonality of the roots of unity and we can express Xi of n as the sum for n that goes from zero to capital N minus 1 of e_j 2Pi over capital N times m times n normalized by capital N. With this, we're ready to compute the expression for A of z. Because now A of z is the standard z transform of the product between Xi of n and x of n. So we replace in the expression for the z transform the value of Xi of n using the roots of unity and we have a double summation. The external summation is over these first terms and the z transform will be computed for the original sequence, and we collect the exponents and we bring this phase term here together with z to obtain one over n that multiplies the sum of capital N copies of the original z-transform of the input sequence computed for different phase of sets of z. So we're taking z, we're multiplying this by pure phase factor, and computing z transform and then summing all these copies together. So A of z, we're just an intermediate quantity that we needed to compute the actual z transform of a downsample sequence as A of z computed in z to the power 1 over n. So the expression for the z-transform is this one, but we're more interested in the restriction of the z-transform to the unit circle because that gives us the spectrum of the downsampled sequence. The expression for that is the following. There's one over N normalization factor in front, that multiplies the sum of capital N copies of the original spectrum, where each copy has been shifted by a multiple of 2Pi over capital N, and where the frequency axis has been stretched out by a factor of capital N. So the interval minus Pi over capital N Pi over N becomes the interval minus Pi, Pi. Let's look at a graphical example, where we're downsampling by two. On the top panel, we have the original spectrum of the input sequence. In the middle panel, we will compute the auxiliary z transform A of z restricted to the unit circle. So remember, this will contain two terms, two shifted copies of the same spectrum, one shifted by 2Pi over 2 times zero. So it will be centered at zero and this is the original copy. Then you will have another copy centering 2Pi over 2 times 2, which is equal to Pi. So here's the second copy of the original spectrum shifted by Pi. Note that like before, in the second panel, we're showing the spectrum with explicit 2Pi periodicity. So we're using a larger support, the minus Pi, Pi. Finally, we have this expansion of the frequency axis, where the minus Pi over 2 to Pi over 2 interval will be mapped to minus Pi, Pi, and so this is the resulting spectrum. So we have started with the spectrum that was half band, and because the spectrum was half band, we had the sufficient gap in the periodic replica of the spectrum to fit another copy of the spectrum without overlap and without aliasing. So this downsampling by two operation actually does not result in irreversible loss of information because we have enough room between non-zero parts of the spectrum to fit another copy of the spectrum in their snugly without overlap, which actually means that the original sequence had been sampled at the sampling frequency which was at least twice the minimum sampling frequency stipulated by the sampling theorem. So by downsampling it, we're bringing it back implicitly to the critical sampling rate. Let's see what happens when this is not the case. This is for instance, a spectrum that goes over 2-3 Pi over 4. So when we build this intermediate quantity A of z restricted to a unit circle, we have one copy centered in zero. We'll still downsampling by two. So there will be a second copy and now centered in pi, and of course, this will overlap with the previous copy of the spectrum. The sum of the two will be like so indicated by the red line, and then the remapping or the stretching of the frequency axis will result in a spectrum for the downsampled sequence that is like so. So you see here this parts and these parts are affected by aliasing. What we can do in discrete time like we did in continuous time is prevent aliasing by first filtering the original signal so that its spectral occupancy is small enough so that there will be no overlapping between the copies, and it's very easy to see that, for instance, in the case of downsampling by two, what we require of the signal is to be band-limited indiscreet time to Pi over 2. So what we can use is an ideal low-pass filter with cut off Pi over two. This will remove of course some information here, but in a controlled fashion so that we're not victims of aliasing, but simply of a trade-off to reduce the sampling rate. With this filtering operation, now the overlap will not occur, and the copies will sit neatly side-by-side, and finally, the resulting spectrum of the downsample sequence will look like so. In general to avoid aliasing, we have to filter the input sequence with a low-pass filter with cutoff frequency Pi over capital N, where N is the down-sampling factor that will follow. Downsampling and upsampling changed the implicit sampling rate by an integer factor, but we can combine them so that the resulting sampling rate change will be an arbitrary fractional number. The combination always goes in the sense that we first upsample the signal, which is the operation that does not change the information content of the original sequence and then we perform downsampling. In doing so, we can combine the filters associated to the interpolation for upsampling. So if we have an upsampler, followed by an interpolating low-pass filter, we know that this lowpass filter will be a low-pass with cutoff frequency Pi over N, and then the downsampler will be preceded by an anti-aliasing filter with low-pass cut off frequency Pi over M. So we can combine these two filters because they would act as a cascade, and since they're both low-pass filters, the smallest cutoff frequency will win, and so we will just use a simple low-pass filter where the cutoff frequencies, the minimum between Pi over N and Pi over M. A practical example that requires a rational sampling rate change is the conversion format between the DVD audio and CD audio. In compact discs, the nominal sampling frequency is 44.1 kilohertz, whereas in DVD audio, the nominal sampling frequency is 48 kilohertz. If we take the ratio of these two sampling frequencies and simplify the fraction, we get a ratio of 160 over 147. These are rather large co-prime terms and they are so biodesign. The reason is DVD audio follows the specification of an earlier standard for digital audio tapes. Digital audio tapes were the first consumer device that allowed people to make perfect digital copies of a medium. So you could go from tape to tape without loss of quality. If the sampling rate of digital audio tapes had been the same as that of CDs, people could have made exactly digital copies of CDs and the industry was obviously afraid of this. So a decision was made to make the sampling frequency of digital audio tapes close to that of CDs, but sufficiently inconvenient so that you could not do a direct copy and you could not use a simple sample rate conversion to move from one standard to the other. But as we will see in the next module, there are clever ways to circumvent this limitation and change from CD to DVD sampling rates without using externally high rate of upsamplers and downsamplers.