I’ve sketched it here for n equals And the angle of H of z will be the angle of this vector minus the angle of this vector. The period of this propetries is from 0 to 2pi, whereas this one wiggles faster, depending on the value of capital N.
There’s only an effect on the angle. So let’s consider x of z to have a factor either in the numerator and denominator of the form z minus z0. And a second is that the coefficients in the numerator polynomial are exactly the same as the coefficients on the right hand side of the difference equation. So if we’re interested in, say, the magnitude of the frequency response, any poles or zeros at the origin, of course, pxf no effect on the magnitude.
It should be clear then, incidentally, that if I give you a system function that’s a rational function of z, that you could construct, in a straightforward way, the difference equation that characterizes that system because you can pick the coefficients off from the numerator.
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Incidentally, it usually is the case– or it is for me, anyway– that often, it’s possible to get a rough picture of what the magnitude of the frequency response is like by looking at this geometric picture. And then, we can use all those other rules of regions of convergences to allow us to figure out from there how far on both sides of the unit circle the region of convergence extends.
These are complex conjugate poles with a radius equal to some value, say r, and an angular spacing equal to omega 0.
And the vector z corresponding to the complex number z minus a is the vector z or z1 minus the vector a. The 0 vector for this example, or in fact any vector pddf the origin out to the unit circle, as we vary omega and, therefore, trace around the unit circle, this downloac doesn’t change in length.
Modify, remix, and reuse just remember to cite OCW as the source. But this is only to illustrate the style of proving properties.
So roughly, we can get a geometric picture of the frequency response for a boxcar sequence by looking at the location of the poles and zeros in the z-plane and the behavior of the vectors as we travel around the unit circle. That’s a general statement. And consequently, let me just show you this function plotted out a little more precisely than I would be able to do at the board. And the result that we get by considering just a simple pole or 0 will, of course, generalize to all the poles and zeros.
And let me begin by reminding you of one of the important properties of the Z-transform for a linear shift invariant system, namely the fact that if we have a linear shift invariant system with a unit sample response, H of n, input x of n, and output y of n, then of course, y of n is the convolution of H of n with x of n.
Properties of Z-Transform
The closer the pole is to the unit circle, the sharper this resonant peak will be and the less damping on the sinusoid. But did I say implicitly what ttansform is? So in general, of course, if a is complex, the poles can have a little movement that way and also a little movement that way. So we have then the statement as I’ve just made it, which we would now like to apply to the interpretation or the generation of the frequency response of a system.
The resulting effect on the Z-transform is to replace z by 1 over z. There are a lot.
The proofs of properties tend to all be in somewhat of a similar vein, as a matter of fact. There’s another important point that we can observe geometrically.
And z1 can be any place in the z transorm. The pole vectors, first of all, introduce only a linear phase term and have no effect on the magnitude. One effect– if we think of x of z as a product of zeros divided by a product of poles– one effect is that there is a constant that, perhaps, collects out in front. And x of n minus k will give us z to the minus k times x of z.
Well, if a is pure imaginary, then the magnitude of a is– I’m sorry. Well, this concludes our discussion of the Z-transform. And that’s a vector going from the origin propfrties to this point. So let’s look at this and for this example see if we can get, for this example, a rough idea of what the frequency response should look like. In the continuous time case, it’s the vertical, or j omega axis in the s-plane that we’re looking at. That is, a useful notion or a useful fact to have, again, stored away in your hip pocket is the effect on the poles and zeros of a system function or a Z-transform of multiplying the sequence by an exponential– maybe a complex exponential, maybe a real exponential.