 The first of the four ideal piecewise constant filters is the ideal low pass filter and the ideal low pass filter would have a frequency response which has 0 phase. In fact, all the filters ideally would have a 0 phase response and a magnitude response that looks like this 1 between minus omega c and plus omega c and 0 else. Now this is very clearly an illustration of why we need to talk about defined almost everywhere. If you look at this response it is precisely specified for all points between minus pi and pi except at the points omega c and minus omega c. At the points omega c and minus omega c the responses are discontinuous, the responses discontinuous. So it is it cannot be specified the right limit in the left limit are not equal. So at that point it is not really correct to talk about the value of the magnitude of the value of the response. Therefore the frequency response exists at all points except at omega c and minus omega c. And therefore we say that for this ideal filter the frequency response whatever it is I mean by whatever it is I mean there is an impulse response that would possibly correspond to this frequency response. The discrete time Fourier transform of that impulse response is defined almost everywhere except for the points omega c and minus omega c. At those 2 points it does not converge and that is why we need to put the almost everywhere. Anyway so this is the ideal low pass filter. Correspondingly we can have now you know this also tells us why we are calling it piecewise constant. The ideal response is constant between minus omega c and plus omega c, constant between omega c and pi, constant between minus pi and minus omega c. Here it is 0 ideally, here it is 0 ideally and here it is 1 ideally in magnitude. On pieces you can divide the frequency axis between minus pi and pi into a finite number of pieces. And on each of these pieces the response is a constant that is why we are calling it piecewise constant. So the ideal response is that we are trying to meet are all piecewise constant. Let us look at the other piecewise constant responses that we would try to design. The ideal high pass filter, high and low remember is only between 0 and pi. So here the ideal high pass filter would ideally take a magnitude of 1 from some omega c to pi and from minus pi to minus omega c and 0 else. So this is the ideal high pass filter. Of course it is 0 phase and this is the magnitude. So here again you notice it is piecewise constant. It is constant on the 3 pieces minus pi to minus omega c, minus omega c to plus omega c and plus omega c to plus pi. We then have the ideal band pass filter and that is easy again. There are 2 points, so omega c 1 and omega c 2 between those 2 points the response is a 1 and otherwise it is 0 and of course once again it is 0 phase. So here again it is piecewise constant. There are exactly 1, 2, 3, 4, 5, 5 pieces on which the response is a constant. Finally we have an ideal band stop filter and as the name suggests this filter would stop a band and pass the rest. It would pass the rest and stop a band. It would stop the band between omega c 1 and omega c 2 and pass the rest and here again this is 0 phase and it is piecewise constant on 5 pieces this one, this one, this one, this one and this one. It is 1 on these 3 pieces and 0 on these 2 pieces. Now we shall also note that it is possible to obtain the ideal impulse response corresponding to each of these ideal filters and we shall illustrate that by starting with the low pass filter. In fact we have already done that job. We do not need to repeat it. We have already calculated the ideal impulse response of the low pass filter. We have already calculated the inverse DTFT of this for the ideal low pass response and we can straight away write that down. Is that right? And the ideal low pass impulse response is equal to sin omega c n divided by pi n whenever n is not equal to 0 and omega c by pi when n is equal to 0. We want to recalculate it back.