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Questions and Answers
What domain is represented by a sequence of values at discrete points in time?
What domain is represented by a sequence of values at discrete points in time?
- Continuous time domain
- Frequency domain
- Discrete time domain (correct)
- Z-domain
The Z-transform converts a signal from the continuous time domain to the Z-domain.
The Z-transform converts a signal from the continuous time domain to the Z-domain.
False (B)
What is the Z-transform of the unit step function, where $x[n] = 1$ for $n \geq 0$?
What is the Z-transform of the unit step function, where $x[n] = 1$ for $n \geq 0$?
z/(z-1)
In the context of systems, what does a 'one sample delay' operation do to the output $y[n]$ relative to the input $u[n]$?
In the context of systems, what does a 'one sample delay' operation do to the output $y[n]$ relative to the input $u[n]$?
A system described as an 'Integrator' performs a summation of the input signal over time, resulting in an output that represents the ______ accumulation of past inputs.
A system described as an 'Integrator' performs a summation of the input signal over time, resulting in an output that represents the ______ accumulation of past inputs.
Match the system operation with its corresponding Z-transform representation.
Match the system operation with its corresponding Z-transform representation.
If a system is described by the equation $y[n] + 0.5y[n-1] = u[n]$, what is the transfer function $Y(z)/U(z)$ of the system in the Z-domain?
If a system is described by the equation $y[n] + 0.5y[n-1] = u[n]$, what is the transfer function $Y(z)/U(z)$ of the system in the Z-domain?
The Discrete-Time Fourier Transform (DTFT) transforms a signal from the Z-domain to the frequency domain.
The Discrete-Time Fourier Transform (DTFT) transforms a signal from the Z-domain to the frequency domain.
Write the formula that represents the relationship between the Discrete-Time Fourier Transform $X(w)$ and the Z-transform $X(z)$ for a discrete signal $x[n]$.
Write the formula that represents the relationship between the Discrete-Time Fourier Transform $X(w)$ and the Z-transform $X(z)$ for a discrete signal $x[n]$.
For a finite-length signal, what concern is typically eliminated when computing its Z-transform?
For a finite-length signal, what concern is typically eliminated when computing its Z-transform?
Given a finite length signal, the Z-transform is a finite ______, meaning that convergence issues are not a concern.
Given a finite length signal, the Z-transform is a finite ______, meaning that convergence issues are not a concern.
Consider the signal $x[k] = {1, 3, 2, -1}$ for $k = {-1, 0, 1, 2}$, respectively. What is the Z-transform $X(z)$ of this signal?
Consider the signal $x[k] = {1, 3, 2, -1}$ for $k = {-1, 0, 1, 2}$, respectively. What is the Z-transform $X(z)$ of this signal?
The Z-transform of the sequence $x[k] = {1,3,2,-1}$ for $k = {-1, 0, 1, 2}$ is expressed as $Z^{-1}+3+2Z-Z^{2}$
The Z-transform of the sequence $x[k] = {1,3,2,-1}$ for $k = {-1, 0, 1, 2}$ is expressed as $Z^{-1}+3+2Z-Z^{2}$
Determine the expression for DTFT, $X(\Omega)$, if $X(z) = z + 3 + 2\cdot z^{-1} - z^{-2}$ .
Determine the expression for DTFT, $X(\Omega)$, if $X(z) = z + 3 + 2\cdot z^{-1} - z^{-2}$ .
In the expression $X(\Omega) = e^{j\Omega} + 3 + 2e^{-j\Omega} - e^{-2j\Omega}$, what substitution was made to derive it from the Z-transform $X(z)$?
In the expression $X(\Omega) = e^{j\Omega} + 3 + 2e^{-j\Omega} - e^{-2j\Omega}$, what substitution was made to derive it from the Z-transform $X(z)$?
Given $x[k] = {1, 3, 2, }$ for $k = {, 0, 1, 2}$, the term corresponding to $k=1$ in the Z-transform $X(z)$ is expressed as $2z^{blank}$
Given $x[k] = {1, 3, 2, }$ for $k = {, 0, 1, 2}$, the term corresponding to $k=1$ in the Z-transform $X(z)$ is expressed as $2z^{blank}$
Match the operation with its Z-transform effect.
Match the operation with its Z-transform effect.
Which property allows us to analyze systems by transforming differential equations into algebraic equations?
Which property allows us to analyze systems by transforming differential equations into algebraic equations?
The Z-transform of $x[n-1]$ is $zX(z)$, where $X(z)$ is the Z-transform of $x[n]$.
The Z-transform of $x[n-1]$ is $zX(z)$, where $X(z)$ is the Z-transform of $x[n]$.
If the Z-transform of $x[n]$ is $X(z)$, what is the Z-transform of $x[n+k]$, where k is an integer?
If the Z-transform of $x[n]$ is $X(z)$, what is the Z-transform of $x[n+k]$, where k is an integer?
Flashcards
Z-Transform Definition
Z-Transform Definition
A mathematical transform used to analyze discrete-time signals and systems. It converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
Z-Transform Formula
Z-Transform Formula
The formula to transform a discrete time domain signal x[n] into the z-domain.
System of 1
System of 1
A system where the output at any time depends only on the input at that same time; mathematically represented as y[n] = u[n].
One sample delay
One sample delay
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Integrator Definition
Integrator Definition
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Discrete-Time Fourier Transform
Discrete-Time Fourier Transform
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DTFT as Z-Transform
DTFT as Z-Transform
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Finite Length Signal
Finite Length Signal
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DTFT from Z-Transform
DTFT from Z-Transform
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Study Notes
- The Z Transform is used in discrete time domain analysis.
- The Z Transform equation is: X(z) = sum(x[n] * z^(-n)) from n = -infinity to infinity.
- The Z Transform is useful in analyzing and manipulating discrete-time signals and systems.
System Analysis using Z-Transform:
- Considering a discrete-time system where u[n] is Input and y[n] is the output.
- If the system's output y[n] is same as the input u[n], Y(z) = 1.
System with One Sample Delay:
- For a system delaying the input by one sample, the output y[n] is a delayed version of the input u[n].
- The Z-transform of the output can be expressed as Y(z) = z^(-1), or 1/z
Integrator System
- If the system acts as an integrator, the output y[n] is the cumulative sum of the input u[n] up to time n.
- For the integrator system, Y(z) = z / (z - 1).
System Described by Difference Equation:
- Consider a system described by the difference equation: y[n] + 0.5y[n-1] = u[n].
- Taking the Z-transform of the entire equation: Y(z) + 0.5z^(-1)Y(z) = U(z).
- The system's transfer function Y(z)/U(z) = z / (z + 0.5).
Discrete-Time Fourier Transform vs. Z Transform
- Discrete-Time Fourier Transform: X(ω) = sum(x[n] * e^(-jωn)) from n = -infinity to infinity.
- Z-Transform: X(z) = sum(x[n] * z^(-n)) from n = -infinity to infinity.
Z-Transform Example
- Find the Z-Transform X(z) for x[k] = {1, 3, 2, -1} for k = {-1, 0, 1, 2}.
- Finite length signal, where X(z) = z + 3 + 2z^(-1) - z^(-2).
- Find the DTFT of x[Ω]. X(Ω) = e^(jΩ) + 3 + 2e^(-jΩ) - e^(-2jΩ)
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