From the transfer function to the frequency response
We have seen that for LTI systems whose output is a weighted sum of a finite number of past inputs and outputs, the transfer function amounts to being a rational function of the form
$H(z)=\frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + \cdots b_M z^{-M}}{1+a_1 z^{-1} + a_2 z^{-2} + \cdots + a_Nz^{-N}}$or equivalently,
$\frac{z^{-M}b_0(z-\zeta_1)(z-\zeta_2)\cdots(z-\zeta_M)}{z^{-N}(z-p_1)(z-p_2)\cdots(z-p_N)}$One of the benefits in expressing the transfer function as the products of those single order polynomials is the elegant way it leads to understanding the magnitude of the transfer function:
$|H(z)|=\frac{|z^{-M}||b_0||(z-\zeta_1)||(z-\zeta_2)|\cdots|(z-\zeta_M)|}{|z^{-N}||(z-p_1)||(z-p_2)|\cdots|(z-p_N)|}$Note what terms like $|z-p|$ represent. The magnitude is $\sqrt{(Re(z)-Re(p))^2+(Im(z)-Im(p))^2}$ which is simply the euclidean distance between $z$ and $p$ on the complex plane. So the magnitude of the transfer function at some point $z$ essentially amounts to being the products of the distances between $z$ and the system's zeros, divided by the products of the distances between $z$ and the system's poles. The closer $z$ is to a zero, the smaller the magnitude of the transfer function will be; the closer $z$ is to a pole, the larger the magnitude of the transfer function will be.
Now recall that a system's frequency response, i.e. the amount the system scales the input at particular frequencies, is simply the z-transform evaluated on the unit circle, for $H(e^{j\omega})=H(z)|_{z=e^{j\omega}}$. If we have a pole-zero plot, we can quickly visualize the magnitude of the system's frequency response by traversing around the unit circle. As we go, the closer we are to a zero, the smaller it will be, while the closer we are to a pole, the larger it will be. As a result of this phenomenon (and the fact that the transfer function is determined by the pole and zero locations), we can see that the locations of a system's poles and zeros in the z-plane totally characterize a system (within a scaling factor). Thus, knowing the poles and zeros of a system is essentially equivalent to knowing exactly what kind of system it is.
Visualizing the magnitude of the frequency response
Consider the pole-zero plot from the example above, with a unit circle overlaid:
CAPTION. The frequency response of this system is simply value of the transfer function, evaluated along this unit circle. The magnitude of the transfer function is the product of the distances to the zeros, divided by the product of the distances to the poles. By traversing along the unit circle and noting these distances, we can visualize what the magnitude of the frequency response will be.
We will start at $\omega=0$. Here we can easily find the value of the magnitude, since $H(e^{j0})=H(1)$. From the transfer function we found in the first example, we have $H(1)=\frac{(1+3+\frac{11}{4}+\frac{3}{4})}{(1-1+\frac{1}{2})}=15$. So $|H(e^{j\omega})$ at $\omega=0$ is $15$. Now, as we move counter-clockwise around the circle (i.e., from $\omega=0$ to $\omega=\pi/2$ to $\omega=\pi$), note what happens. As $\omega$ approaches $\pi/4$ on the unit circle, it is drawing nearer and nearer to a pole; therefore, the magnitude of the frequency response is going to increase as $\omega$ progresses from $0$ to $\pi/4$. At that point, we will be travelling away from that pole, and nearer to the three zeros; accordingly, the magnitude of the frequency response will decrease. Eventually we will actually land squarely on a zero at $\omega=\pi$; this is obviously as close as we could possibly get to a zero. As the distance to the zero at that point is zero, the magnitude of the frequency response there is zero. We can repeat this process moving clockwise from $\omega=0$ to $\omega=-\pi$ and the magnitude will change in the precisely same way, due to the symmetry. Having moved along the entire unit circle, we can see how our intuition matches what is indeed the magnitude of the frequency response, found via plotting software:
CAPTION. Note that the magnitude is $15$ at $\omega=0$, then increases to a peak at about $\pm\pi/4$, and then decreases to $0$ as $\omega$ traverses from $\pm\pi/4$ to $\pm\pi$.
The transfer function roc, and stability
It has been explained previously that the region of convergence of a z-transform are the $z$ for which the sum $\sum x[n] z^{-n}$ converges. There is an alternative, though related definition that is also often used in signal processing, that the region of convergence are $z$ for which the sum $\sum x[n]z^{-n}$ converges ABSOLUTELY, i.e. that $\sum |x[n] z^{-n}|$. For virtually all of the signals that we will be considering, those two definitions are equivalent. The reason the second definition is often used is because of a relationship it establishes with the stability of the system.
Suppose that a system's transfer function $H(z)$ has a particular ROC (defined in the absolutely summable sense). If the unit circle ($|z|=1$) is contained by this ROC, then the system is BIBO stable. To prove this fact, we see simply that the $|z|=1$ being in the ROC means that $\sum |h[n]e^{j\omega}|$ converges for all $\omega$, and then note that for $\omega=0$, the convergent sum is $\sum |h[n]|$. Previously we proved that for LTI systems, $\sum |h[n]|$ converging is equivalent to the system being BIBO stable.
So if the unit circle is in a transfer function's ROC, then that system is BIBO stable. This is a helpful way to determine stability. We do not need to prove that an arbitrary bounded input will produce a bounded output, nor find the impulse response and determine its absolute summability; we simply need to know the transfer function's ROC.
If the system in question is causal (which is the case for most all practical systems of interest), then the ROC will extend outwards from the outermost pole. This means that, for causal systems, the location of the poles also determine BIBO stability. If all of a causal system's poles lie within the unit circle, then the ROC will extend to include the unit circle, and hence the system will be BIBO stable.
Stability and the pole/zero plot
Consider again the pole/zero plot from the system in the previous examples:
CAPTION. Since the ROC contains the $|z|=1$ unit circle, this system is BIBO stable.
Poles/zeros and fir/iir systems
We have seen that the poles and zeros of a system characterize it, which of course includes the nature of its frequency response and whether or not the system is BIBO stable. Given how significant the poles and zeros of a system are, it should come as no surprise that knowing about the poles and zeros also allows us to determine whether the system is FIR or IIR.
Put simply, if a system's transfer function has any poles (other than at $z=0$), then the system is IIR. If it has only zeros, then it is FIR. This follows from what we saw with regard to the nature of different kinds of ROCs. If the ROC is the entire $z$ plane (with the possible exception of $z=0$), then the impulse response is of finite duration; hence the system is FIR. But if there are any non-zero poles on the plane, than the ROC will either be a disk, an annulus, or will extend outside of a disk; in each of those cases, the impulse response is of infinite duration.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
