Method for representing DT signals as superpositions of complex geometric (exponential) functions.
Lecture #15:
THE BILATERAL Z-TRANSFORM
Motivation: Method for representing DT signals as superpositions of complex geometric (exponential) functions
Outline:
Review of last lecture
The bilateral Z-transform
– Definition
– Properties
Inventory of transform pairs
Conclusion
Review of last lecture
Solve linear difference equation for a causal exponential input
Solve homogeneous equation for n>0
Solve characteristic polynomial for λ.
Solve for a particular solution for n>0
Assuming
and solving for Y yields
Logic for an analysis method for DT LTI systems
characterizes system compute
efficiently.
In steady state, response to
is
.
Represent arbitrary x[n] as superpositions of
on z.
Compute response y[n] as superpositions of
on z.
I. THE BILATERAL Z-TRANSFORM
1/ Definition
The bilateral Z-transform is defined by the analysis formula
is defined for a region in z — called the region of convergence — for which the sum exists.
The inverse transform is defined by the synthesis formula
Since z is a complex quantity,
is a complex function of a complex variable. Hence, the synthesis formula involves integration in the complex z domain. We shall not perform this integration in this subject. The synthesis formula will be used only to prove theorems and not to compute time functions directly.
a/ Approach
An inventory of time functions and their Z-transforms will be developed by
Using the Z-transform properties,
Determining the Z-transforms of elementary DT time functions,
Combining the results of the above two items.
b/ Notation
We shall use two useful notations — Z{x[n]} signifies the Z-transform of x[n]and a Z-transform pair is indicated by
2/ Properties
a/ Linearity
The proof follows from the definition of the Z-transform as a sum.
b/ Delay by k
This result can be seen using the synthesis formula,
c/ Multiply by n
This result can be seen using the analysis formula.
Most proofs of Z-transform properties are simple. Some of the important properties are summarized here.
R, R1, and R2 are the ROCs of
,
, and
, respectively. * Exceptions may occur at z = 0 and z = ∞.
II. Z-TRANSFORMS OF SIMPLE TIME FUNCTIONS
1/ Unit sample function
The Z-transform of the unit sample is
for all values of z, i.e., the ROC is the entire z plane.
2/ Unit step function
The unit step and unit sample functions are simply related.
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?
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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
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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?