# input substituition

Explain why a change in a firm's total fixed cost of production will shift its average total cost curve, but not its marginal cost curve.

Provide two example of input substituition.

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#### Solution Preview

The simplest way to see why changes in a firm's total fixed cost of production shifts average total cost curve but not the marginal cost curve is to look at their definitions.

ATC = Total Cost / Quantity = (Total Fixed Cost + Total Variable Cost)/Quantity = (Total Fixed Cost / Quantity) + (Total Variable Cost / Quantity)

MC = Change in Total Cost / Change in Quantity = (Change in Total Fixed Cost + Change in Total Variable Cost)/Change in Quantity

As output increases the fixed cost is still fixed, and there is no change in total fixed cost. Thus the equation becomes

MC = Change in Total Cost / Change in Quantity = (Change in Total Fixed Cost / Change in Quantity) + (Change in Total Variable Cost / Change in Quantity) = 0 + (Change in Total Variable Cost / Change in Quantity) = (Change in Total Variable Cost / Change in Quantity)

Thus fixed cost ...

#### Solution Summary

The input substituition is examined.

Discrete Time Systems

1. a) Find the z-transform of the following system transfer function assuming the input is a unit staircase (i.e. zero order held). The sampling rate is T=0.2.

P(s) = (s+2)/(s+1)(s+5)

b) Given the z-transform of a sequence, U(z) = Z{u},

U(z) = z/(z^2 - 0.8z + 0.6)

i. Find the first 4 terms of u by long division.

ii. Find the underlying signal u(t) assuming that T=1 and no aliasing has occured.

2. a) The system (P_s)(s) = 4/(s^2 + 2s + 3) is subject to a staricase input with T = 0.2 seconds. Calculate the Z domain transfer function (P_z)(z).

b) The system, (P_s)(s) = 1/(s + 2) is subject to a staircase input with T = 0.1 seconds. Calculate the w-domain transfer function (P_z)(w) and comment on the relationship between s- and w-doman transfer functions.

3. a) Given (P_s)(s) = 4/(s + 3), find the w domain description for T = 0.2 and comment on the similarity and difference between (P_s)(s) and (P_w)(w). Use 3 decimal place in your calculations.

bi) Calculate the z-transform of the signal, y)t_ = e^(0t)sin(5t) sampled at T = 1.0.

bii) By finding the inverse z-transform, show that aliasing has occurred and explain why.

c) Find the first 4 terms in the inverse z-transform of Y(z) = 2z/(z^2 + 0.5z + 0.3) by means of long division. Use the final value of the signal to confirm that your results make sense.

4. a) The system, (P_s)(s) = 3/(s^2 + s + 2.5) is subject to a staircase input with T=0.2. Calculate the Z doman transfer function (P_z)(z).

b) A sequence, y_B, has a Z transform,

Y(z) = (z^2 + z)/(z^3 + 2z^2 + 1.4z + 0.3)

Find the first 5 terms in the sequence (y_0 ... y_4) by long division.

c) By considering the initial and final value theorems in the Z domain, find equivalent initial and final value theorems in the W domain and confirm that they are similar to the S domain theorems.

5. Show that aliasing will occur if the signal y(t) = e^-t(sin)(2t) is sampled with T=0.2 by finding the z transform of the sampled sequence and then finding the inverse z transform.

c) If a system has the transfer function P(s) = 1/(s/(3+1)) and T = 0.2, find (P_z)(w).

d) Show that the unit step signal, u(t) = sigma(t)

6. ai) The system, (P_s)(s) = (4s + 7)/(s^2 + 3s + 2) is subject to a staircase input with T = 0.5. Calculate the z domain transfer function, (P_z)(z).

aii) Use the final value theorem to find the final value of the output if the input is a step. Does your answer make sense with respect to the final value of the step response of the continuos system?

b) Given (P_s)(s) = 2/(s + 3), find the w domain description for T = 0.2 and comment of the similarity and difference between (P_s)(s) and (P_w)(w).

c) Given (P_z)(z) = (2z^2 - 0.5z + 1)/(z^2 + 0.2z + 0.5) and u_o = 1, u_1 = 2, u_n = 0, n =/ 0, 1, initioal y = 0.

i) Calculate y_n for n=0, 1, 2, and 3 by writing the difference equation and substituition.

ii) Does the oscillating behaviour make sense in terms of the pole locations in the z plane?

7. a) Find the z-transform of the following system transfer function assuming the input is a unit staircase (i.e. zero order held). The sampling rate is T = 0.2.

P(s) = 3/(s^2 + s + 1)

b) A signal y(t) = e^(-0.t)sin*2pi(t) is sampled at a rate T = 0.5, Show that the z-transform of the sampled sequence is Y(z) = (ze^-0.25)(sin(pi))/(z + e^-0.5)^2 = 0. Make a sketch to explain what has happened.

c) Given P(s) = 1/(s+1), find (P_z)(w) for T = 0.1.

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