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Questions and Answers
What is the z-transform, like the Laplace transform, used for?
What is the z-transform, like the Laplace transform, used for?
design, analysis and monitoring of systems
What is the z-transform considered to be?
What is the z-transform considered to be?
the discrete-time part of the Laplace transform and a generalization of the Fourier transform of a sampled signal
A working knowledge of the z-transform is essential to the study of digital filters and systems?
A working knowledge of the z-transform is essential to the study of digital filters and systems?
True
What does the z-transform allow insight into?
What does the z-transform allow insight into?
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What does the z-transform enable one to obtain?
What does the z-transform enable one to obtain?
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DTFT can be applied to unstable LTI systems?
DTFT can be applied to unstable LTI systems?
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The z-transform provides an easier way to test stability and causality?
The z-transform provides an easier way to test stability and causality?
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In the context of time domain signals, what does the convolution of two signals translate to?
In the context of time domain signals, what does the convolution of two signals translate to?
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What is another benefit of using the z-transform?
What is another benefit of using the z-transform?
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What is X(z) defined as?
What is X(z) defined as?
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The complex variable 's' is defined as σ − jω?
The complex variable 's' is defined as σ − jω?
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What is the Laplace transform used for?
What is the Laplace transform used for?
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The Z-transform was derived from the Fourier transform?
The Z-transform was derived from the Fourier transform?
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The z-transform for causal system is defined as: X(z) = Σ x(n)z^-n, from n = -∞ to ∞?
The z-transform for causal system is defined as: X(z) = Σ x(n)z^-n, from n = -∞ to ∞?
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Poles and zeros in z-plane are denoted by 'o' and 'x' respectively?
Poles and zeros in z-plane are denoted by 'o' and 'x' respectively?
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The unit circle in z-plane corresponds to the jω axis of the s-plane.
The unit circle in z-plane corresponds to the jω axis of the s-plane.
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What is the point z = re^jθ a vector of?
What is the point z = re^jθ a vector of?
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The contour |z| = 1 is a circle with unity radius on the z-plane, referred to as the unit circle.
The contour |z| = 1 is a circle with unity radius on the z-plane, referred to as the unit circle.
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What is the DTFT given by?
What is the DTFT given by?
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X(e^jω) is a complex valued discontinuous function.
X(e^jω) is a complex valued discontinuous function.
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What does 'f' represent?
What does 'f' represent?
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The relation between Z-transform and DTFT is that DTFT is to evaluate z-transform on a unit circle?
The relation between Z-transform and DTFT is that DTFT is to evaluate z-transform on a unit circle?
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What does the ROC of X(z) represent?
What does the ROC of X(z) represent?
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The ROC cannot include any poles?
The ROC cannot include any poles?
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The ROC for finite duration sequences is the entire z-plane except possibly z = 0 or z = ∞?
The ROC for finite duration sequences is the entire z-plane except possibly z = 0 or z = ∞?
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Where does the ROC extend outward from for right sided sequences?
Where does the ROC extend outward from for right sided sequences?
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Where does the ROC extend inward from for left sided sequences?
Where does the ROC extend inward from for left sided sequences?
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What is the ROC for X1(z)?
What is the ROC for X1(z)?
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What is the ROC of Z{δ(n)}?
What is the ROC of Z{δ(n)}?
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The expression for X(z) in terms of a and z can be used to find the inverse ZT using partial fractions.
The expression for X(z) in terms of a and z can be used to find the inverse ZT using partial fractions.
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The form of X(z) in terms of z and a can be used to find the pole and zero locations.
The form of X(z) in terms of z and a can be used to find the pole and zero locations.
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If |az^-1|<1, then |z|<|a| ?
If |az^-1|<1, then |z|<|a| ?
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The ROC for a causal sequence is |z|>|a|?
The ROC for a causal sequence is |z|>|a|?
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The linearity property of the z-transform states that the z-transform of a sum of signals is equal to the sum of the z-transforms.
The linearity property of the z-transform states that the z-transform of a sum of signals is equal to the sum of the z-transforms.
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The time shifting property of the z-transform states that the z-transform of a signal shifted by k samples is equal to the z-transform of the original signal multiplied by z^-k.
The time shifting property of the z-transform states that the z-transform of a signal shifted by k samples is equal to the z-transform of the original signal multiplied by z^-k.
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The z-domain differentiation property of the z-transform states that the z-transform of the nth derivative of a signal is equal to the nth derivative of the z-transform.
The z-domain differentiation property of the z-transform states that the z-transform of the nth derivative of a signal is equal to the nth derivative of the z-transform.
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The z-scale property of the z-transform states that the z-transform of a signal scaled by a factor of a is equal to the z-transform of the original signal divided by a.
The z-scale property of the z-transform states that the z-transform of a signal scaled by a factor of a is equal to the z-transform of the original signal divided by a.
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The time reversal property of the z-transform states that the z-transform of a time-reversed signal is equal to the z-transform of the original signal with z replaced by 1/z.
The time reversal property of the z-transform states that the z-transform of a time-reversed signal is equal to the z-transform of the original signal with z replaced by 1/z.
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The convolution property of the z-transform states that the z-transform of the convolution of two signals is equal to the product of their z-transforms.
The convolution property of the z-transform states that the z-transform of the convolution of two signals is equal to the product of their z-transforms.
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The ROC of the convolution property of the z-transform is always the intersection of the individual ROCs?
The ROC of the convolution property of the z-transform is always the intersection of the individual ROCs?
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What is the transfer function H(z) determined by?
What is the transfer function H(z) determined by?
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For a system with a finite impulse response, the transfer function H(z) is given as the sum of all the coefficients b multiplied by z raised to the negative power?
For a system with a finite impulse response, the transfer function H(z) is given as the sum of all the coefficients b multiplied by z raised to the negative power?
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The impulse response of an IIR system extends for -∞≤n≤∞?
The impulse response of an IIR system extends for -∞≤n≤∞?
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The frequency response H(e^jω) can be calculated by substituting z = e^jω in the transfer function H(z)?
The frequency response H(e^jω) can be calculated by substituting z = e^jω in the transfer function H(z)?
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The DTFT and the z-transform are directly related.
The DTFT and the z-transform are directly related.
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Frequency response H(e^jω) is independent of the impulse response h(n)?
Frequency response H(e^jω) is independent of the impulse response h(n)?
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Study Notes
Digital Signal Processing Course
- Course offered by Dr. Manal Shaban, Dr. Nabil Sabour, and Dr. Adel Fathy Khalifa
- Course is part of the Information Technology program at EELU (The Egyptian E-Learning University)
- Course year is 2024-2025, Fall Semester
Z-Transform
- Z-transform is a mathematical tool for system design, analysis, and monitoring, similar to Laplace Transforms
- It's the discrete-time component of the Laplace transform, generalizing the Fourier Transform of a sampled signal
- Crucial for understanding digital filters and systems
- Enables a deeper understanding of discrete-time LTI (Linear Time-Invariant) systems and their interaction with signals, surpassing the capabilities of the DTFT (Discrete-Time Fourier Transform)
DTFT works for stable LTI systems; Z-transform also works for unstable LTI systems
Derivation of Z-Transform
- Derived from Laplace transform
- Replace s (complex variable: σ + jω) with z = esT (where T is the sampling time) in the Laplace transform equation
- This yields a two-sided z-transform
- For a causal system, the transformation simplifies to a one-sided equation
Definition of Z-Transform
- Defined as the sum of the sequence multiplied by powers of
z<sup>-n</sup> - For causal systems, it's the sum from n = 0 to infinity, X(z) = Σx(n)z−n
- For a two-sided equation, sum from n = -∞ to ∞, X(z) = Σx(n)z−n
Geometric Representation of Z-Transform
- Z represents a complex number in the z-plane, often depicted as a circle of radius r
- z = rejω (r is the magnitude, ω is the angle)
- Z-plane poles are denoted with "x" on the plot, and zeroes with "o"
- The unit circle corresponds to the jω-axis in the complex s-plane
- Numerator polynomial root (zeros) and denominator polynomial root (poles) are represented in the visual of the Z-plane
Discrete-Time Fourier Transform (DTFT)
- The DTFT represents a signal in the frequency domain
- The DTFT is related to the z-transform via z=ejω on the unit circle.
ROC (Region of Convergence)
- The set of z values where the Z-Transform is finite
- The ROC is a ring or disc centered at the origin of the z-plane
- The ROC doesn't include any poles
- Finite-duration sequences have ROC covering the entire z-plane, except possibly at 0 or ∞
- Right-sided sequences have ROC extending outward from the outermost pole to infinity.
- Left-sided sequences have ROC extending inward from the innermost pole to zero.
Properties of Z-transform
- Linearity: ax[n] + by[n] ←→ aX(z) + bY(z).
- Time Shifting: x[n-n0] ←→ z-n0*X(z)
- z-Domain Differentiation: nx[n] ←→ zdX(z)/dz
- Z-Scale Property: an*x[n] ←→ X(z/a) .
- Time Reversal: x[-n] ←→ X(z−1)
- Convolution: h[n]*x[n] ←→ H(z)*X(z)
Rational Z-Transform
- For practical signals, Z-transform is expressed as the ratio of two polynomials
- X(z) = G*( (z-z1)(z-z2)…(z-zM) ) / ((z-p1)(z-p2)…(z-pN) )
- G is a scalar gain, zi are the zeros, and pi are the poles.
Z-Transform Function for Linear Time-Invariant (LTI) Systems
- H(z) = Σh(n)z-n is the system function
- Converts the time-domain difference equation (describing the system output) into a z-domain representation
- H(z) = Y(z)/X(z)
Relationship Between Frequency Response and Difference Equation for LTI System
- Establishes the relationship between system behaviour in frequency domain and time domain
- Frequency response is the system's characteristic in the frequency domain.
- H(z)|z=ejω = H(ejω) gives the frequency response by evaluation on the unit circle in the Z-plane
Other Relationships
- Relationships between x(n), y(n) and transfer functions H(z): taking DTFT, Inverse z-Transform and solving for specific variables to obtain the desired output.
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Description
Test your knowledge on the Z-transform, a crucial mathematical tool for system design and analysis in digital signal processing. This quiz covers its derivation from Laplace transforms, applications in digital filters, and understanding of LTI systems. Challenge yourself to deepen your understanding of discrete-time signals.
