Math Problem Statement
Solution
The image contains two questions related to the Z-transform of given sequences, which appear to involve sinusoidal and hyperbolic trigonometric functions.
Let's break down each question.
7) The Z-transform of the sequence is:
The Z-transform for a sinusoidal sequence like can be calculated using the standard result:
In this case, we have and . We need to apply the formula, and based on the choices provided, we can evaluate the correct result by calculating the numerator and denominator step by step.
8) The Z-transform of :
For hyperbolic functions like , the process is similar to sinusoidal functions, but instead of sine and cosine, the hyperbolic counterparts and are involved. The Z-transform can be given by:
Here, , and we can substitute that value into the general formula to simplify it, yielding one of the provided options.
Would you like me to calculate the exact Z-transforms for each of these, or do you need help with a specific step in the process? Let me know if you'd like details!
5 Follow-up questions:
- What are the steps to compute the Z-transform of any sinusoidal function?
- How does the Z-transform handle hyperbolic functions like and ?
- Why does the region of convergence or appear in these problems?
- How do you differentiate between the numerators for sine and cosine Z-transform formulas?
- What changes when you compute the Z-transform for sequences involving phase shifts?
Tip: The Z-transform is especially useful for solving difference equations and analyzing discrete-time systems in signal processing.
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Math Problem Analysis
Mathematical Concepts
Z-transform
Sinusoidal Functions
Hyperbolic Functions
Formulas
Z{sin(ωn + θ)} = (z sin θ - z^2 sin (ω + θ)) / (z^2 - 2z cos(ω) + 1)
Z{cosh(ωn + θ)} = (z cosh θ - cosh ω) / (z^2 - 2z cosh(ω) + 1)
Theorems
Z-transform properties for trigonometric and hyperbolic functions
Suitable Grade Level
Undergraduate level (Calculus/Discrete Mathematics)