ECE 341
Signals and Systems
Lecture 11:
Discrete-Time Analysis using z-Transform
Praveen Sekhar and Aref Majdara, PhD
School of Engineering and Computer Science
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Definition
 z-Transform is the discrete-time equivalent of the Laplace
Transform.
 The bilateral z-Transform:
where z is a complex variable: 𝑧𝑧 = 𝑟𝑟𝑒𝑒 𝑗𝑗𝜃𝜃
 Inverse z-Transform:
Integration in a counter-clockwise direction around a closed path in the complex plane
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Definition
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 Transform pairs:
 Note that,
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Linearity Property
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 The z-transform is a linear operator:
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The Unilateral z-Transform
 Similar to Laplace transform, with the bilateral transform the
inverse transform will not be unique.
o Thus, it is more convenient to use the unilateral transform.
o It deals with causal signals.
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The Region of Convergence (ROC)
 The summation is on 𝑛𝑛: 0 → ∞. So, it may not converge.
o Thus, we need to determine the Region of Convergence (ROC).
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ROC
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The Region of Convergence (ROC)
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 Important note:
o Some signals may have the same z-transform, but different
ROCs.
o In that case, the ROC will be needed, in order to uniquely find
the signal using the inverse z-transform.
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The Inverse z-Transform
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 By definition:
 However, whenever possible, instead of doing the complicated
integration in z-domain, we use Partial Fraction Expansion and the
table of transform pairs.
o Similar to the inverse Laplace transform.
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 To avoid the 𝑢𝑢[𝑛𝑛 − 1] in the answer, we can use the “modified”
partial fraction expansion, instead:
“Modified” Partial Fraction Expansion
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Modified PFE
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Multiply both sides by z and then let 𝑧𝑧 → ∞:
Let 𝑧𝑧 = 0 on both sides:
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System Transfer Function
 For an LTI system, 𝐻𝐻 𝑧𝑧 =
𝑌𝑌[𝑧𝑧]
𝑋𝑋[𝑧𝑧]
 𝐻𝐻(𝑧𝑧) is the z-transform of the unit impulse response ℎ[𝑛𝑛].
 𝐻𝐻(𝑧𝑧) can be used to find the ZSR of the system to any input:
o Input: 𝑥𝑥[𝑛𝑛] ,
𝐻𝐻 𝑧𝑧 =
𝑌𝑌[𝑧𝑧]
𝑋𝑋[𝑧𝑧]
Zero Initial Conditions,
Transfer Function: 𝐻𝐻[𝑧𝑧] ,
𝑦𝑦[𝑛𝑛] = ?
 𝑌𝑌 𝑧𝑧 = 𝐻𝐻 𝑧𝑧 𝑋𝑋[𝑧𝑧]  𝑦𝑦[𝑛𝑛] = 𝒵𝒵 −1 {𝑌𝑌[𝑧𝑧]}
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Properties of the z-Transform
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Solving Linear Difference Equations
using z-Transform
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Solving Difference Equations
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Take the z-Transform of the difference equation.
o Using the Left-Shift and/or Right-Shift properties.
2. Solve the algebraic equation in z-domain.
3. Take the inverse z-transform of the result.
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 The equation is in “advance” form.
 First, convert it to “delay” form.
o By replacing 𝑛𝑛 with 𝑛𝑛 − 2:
 Then, find the difference terms, using:
o For the left side:
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o For the right side:
 𝑥𝑥 𝑛𝑛 is a causal signal, so:
 Then:
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 𝑧𝑧-transform of the difference equation:
 Solve the equation in 𝑧𝑧-domain (to find 𝑌𝑌[𝑧𝑧]):
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 Taking the answer back to the 𝑛𝑛 domain:
(Modified P. F. E.)
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Zero-Input and Zero-State Components
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 In the 𝑧𝑧-domain equation, separate the terms into:
o Terms arising from the initial conditions  Zero-Input Response
o Terms arising from the input  Zero-State Response

From the previous example:
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Zero-Input and Zero-State Components
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 Modified Partial Fractions:
 Finally:
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The Transfer Function
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Transfer Function
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 For an LTID system:
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(Modified P. F. E.)
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Invers Systems
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 If system 𝑆𝑆𝑖𝑖 (with transfer function 𝐻𝐻𝑖𝑖 (𝑧𝑧)) is the inverse of
system 𝑆𝑆 (with transfer function 𝐻𝐻(𝑧𝑧)), then:
o Or equivalently:

Based on the “Time Convolution” property:
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System Stability
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System Stability
 Can be determined from the system transfer function:
𝑃𝑃[𝑧𝑧]
𝐻𝐻 𝑧𝑧 =
𝑄𝑄[𝑧𝑧]
 𝑃𝑃[𝑧𝑧] and 𝑄𝑄[𝑧𝑧] are polynomials of 𝑧𝑧.
 If all poles of 𝐻𝐻[𝑧𝑧] (i.e. roots of 𝑄𝑄[𝑧𝑧]) are within the unit circle,
o  All terms of ℎ[𝑛𝑛] will be decaying exponentials.
o  ℎ[𝑛𝑛] will be “absolutely summable”.
o  The system will be BIBO stable (externally stable).
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