 But now we have you know we have seen something very interesting we can combine LSI systems in cascade of course it is a minor variation or a very easy question to answer when we ask what happens to LSI systems in parallel that is very easy to answer. So suppose you have the same Xn being given to 2 LSI systems in parallel with impulse responses H1n and H2n. By parallel you mean you apply the same input and then add the outputs the answer is very easy can we find an equivalent system for this and if so what is the equivalent system. Well it is very easy to see that y of n is equal to Xn convolved with H1n plus Xn convolved with H2n and of course this can be written summation on k overall k integer H1n minus k plus summation on k Xk H2n minus k which of course is very easy to combine this does not require any great knowledge of just a simple distributivity property of multiplication that we know very well where n is equal to H1n plus H2 very easy. So in fact when we have 2 LSI systems in parallel there is an equivalent LSI system whose impulse response is the sum of the impulse responses of the individual LSI system simple enough. So now we have well in a position to deal with any combination of LSI systems in cascade or in parallel or a combination of cascade in parallel is that right we can always find an equivalent LSI system for them the only problem is if you wish to find an equivalent LSI system for a cascade it is hard work because if you do a very cumbersome operation convolution. So this is another reason why we might want to see if you can go to some other domain or go into some other mode where I can carry this operation out more easily and in fact it is quite beautiful how several questions in signal processing converge to one answer. Now recall that the whole reason why we started discussing linear shift invariant systems was to deal with sinusoids we come back to the same story again. There we had those sinusoids we had complex exponentials because we did not want sinusoids and the reason why we wanted complex exponentials was that although a change of amplitude could be could be represented as multiplication by a constant changes of phase could not but they could if you dealt with complex exponentials and therefore we asked what is the system which leaves a complex exponential as it is in frequency but changes only the amplitude and phase and we said that system must be LSI now we need to justify that. So why have we worked so hard to build all these properties of convolution unless we can show that we have indeed got where we wanted to. So now let us ask the question what happens when we feed a complex exponential of angular frequency omega we will use small omega and let me now bring in some notation here. So what happens when we feed in a complex exponential of angular frequency omega into an LSI system let us call it S with impulse response h n. Now a few remarks about the angular frequency omega hence forth we are going to use what are called normalized periods and frequencies and what we mean by that is that we shall assume at the sampling angular frequency or the well let us first take the sampling frequency we will take the sampling frequency to be 1 unit you know the unit is our choice ultimately whatever it is if it is 10 kilo hertz we will say 10 kilo hertz is 1 unit if it is 1 mega hertz we will say 1 mega hertz is 1 unit. So we will say the sampling frequency is 1 unit is the unit frequency whatever so you can always choose your unit and therefore of course the sampling period is also 1 unit obviously it is the reciprocal of the frequency of course these are units of different quantities sampling frequency is as units of frequency whatever they might be and sampling periods have units of time therefore the sampling angular frequency becomes 2 pi times 1 unit. So whenever we are dealing with a discrete system we will assume that we have chosen the unit so that 1 unit is equal to the sampling frequency and therefore the sampling period is 1 and thereby we shall use small omega to denote what is called the normalized angular frequency or the angular frequency in these units now we need to reflect for a minute what the unit of this normalized angular frequency would be you see when we normalize we divide so for example if you say you have normalized the angular or you have normalized the frequency of sampling you have actually divided the sampling frequency by the actual sampling frequency. So for example if your sampling frequency were 1 mega hertz you have divided all frequencies by 1 mega hertz to get the normalized frequency similarly if you have divided the angular frequency in a similar way to get the normalized quantity then in fact what you have done is to replace radians per second radians per second was the unit of angular frequency as it were but you have divided this by the sampling frequency in actual value. So radians per second divided by per second or hertz leaves you with radians and therefore the units of normalized angular frequency here are radians not radians per second similarly the units of period or units of frequency are null there are no units because you have divided hertz by hertz they are just numbers yes there is a question okay so the question is is it appropriate to think of this as a unit or to think of this as an angle well both are correct you see what we are saying in a way what omega denotes is how much of angle is covered in a unit sample time in a sample time so you see when omega is equal to 2 pi that means when you have some when you have used the sampling frequency itself then you have covered the angle of 2 pi in a sample time right so essentially omega is a measure of the angle covered in one sample time that is another interpretation yes that is why the unit is radians all right then we will agree then to use the normalized angular frequency because you see we are going to use the integer n to denote the sample number and also to denote the sample time since the sampling time is unity so therefore we have e raised the power j omega n as the complex exponential otherwise we would have to write down e raised the power omega ts times n and so on right so we do not need to do that now so when we feed e raised the power j omega n to this LSI system s with impulse response hn we know what we get out we will get out e raised the power j omega n convolved with hn now here we are going to invoke the commutativity of convolution and that would give us hn convolved with e raised the power j omega n equally well from commutativity is that right so if we use that expression then it gives us summation k going from minus to plus infinity hk e raised the power j omega n minus k and that is very easy to break up it is k from minus to plus infinity hk e raised the power j omega n e raised the power minus j omega k now note that here e raised the power j omega n is independent of k so I can draw it out of the summation and I am left then with an infinite summation on k the infinite summation k is in fact a function of omega and a function of h so let me denote this summation summation from k going from minus to plus infinity hk e raised the power minus j omega k by capital h of omega where upon what we have said essentially is that e raised the power j omega n going into the system s with impulse response hn has led to e raised the power j omega n coming out but multiplied by a complex constant capital H of omega please note that capital H of omega is a complex constant but of course at the moment we have kind of you know brushed something very important under the carpet let us look back at this expression here h omega is summation k going from minus to plus infinity hk e raised the power minus j omega k there is an infinite summation here now infinite summation is not guaranteed to converge and infinite summation can diverge in fact this is a problem with convolution in general and we had brushed this issue under the carpet even the last time it conveniently ignored that issue altogether that is because we had dealt with finite length sequences so convergence was never an issue but in case our sequences happen not to be of finite length there could be a problem in the summation whether it is in the context of convolution or it is in the context of this response that we see here.