Z-Transform: Discrete-Time Signal Analysis

Z-Transform

• The objectives are to become familiar with the Z-transform
• The objectives are also to learn to apply the Z-transform for analysis of discrete-time signals, and differentiate between Bilateral and Unilateral Z-transforms

Definition of Z-Transform

• In mathematics and signal processing, Z-transform converts a time domain signal, represented as a sequence of real or complex numbers, into a complex frequency domain representation
• Z-transform can be considered a discrete-time equivalent of the Laplace transform

Time Domain vs Frequency Domain

• Time-domain representation gives the amplitudes of a signal at specific time instances when sampled
• In many instances, knowing the frequency content is important rather than signal amplitudes

Complex Numbers

• A complex number in Cartesian Form is written as z = x + jy
• x represents the real part of z expressed as Rz
• y represents the imaginary part of z expressed as Iz
• j represents the square root of -1 and is used in engineering notation, i is also a square root of -1
• A complex number in polar form is z = re^(jϕ)
• r represents the modulus or magnitude of z
• ϕ represents the angle or phase of z
• exp(jϕ) = cos(ϕ) + j sin(ϕ)
• Complex exponential is defined as e^z = e^(x+jy) = e^x * e^(jy) = e^x(cos y + j sin y)

Laplace Transform

• Laplace Transform converts a function of time into a function of complex frequency

Z-Transform

• A Z-transform also converts a function of time into a function of complex frequency

Continuous and Discrete Signals

• The Laplace transform is used to convert functions from continuous time
• The Z-Transform is used to convert functions from discrete time

Euler's Identity

• Euler's identity is a mathematical equation with 5 fundamental constants
• e^(ix) = cos(x) + isin(x)
• e^(-ix) = cos(-x) + isin(-x)

Laplace and Z-Transforms

• Laplace and Z transforms are tools to convert functions of time into functions of a complex variable
• These transforms analyze systems and solve differential equations in engineering and signal processing

Laplace Transform Details

• Laplace transform converts continuous-time functions into complex-valued functions of a complex variable, "s"
• The Laplace transform solves linear differential equations with constant coefficients

Key Points of Laplace Transform

• The domain of the Laplace transform is continuous-time functions
• The transform variable is "s", a complex number
• Laplace Transform can solve differential equations and analyze systems, it is also used in control theory

Z-Transform Details

• The Z-transform is converts discrete-time functions (sequences) into complex-valued functions of complex variable, typically denoted as "z"
• Z-transforms commonly appear in digital signal processing and control systems

Key Points of Z-Transform

• The domain of the Z-Transform is discrete-time functions, or sequences
• The transform variable is "z", a complex number
• Z-Transforms appear within digital signal processing, control systems and discrete-time systems analysis

Key Differences of Z-Transform and Laplace Transform

• Laplace transform deals with continuous-time functions
• Z-transform deals with discrete-time functions
• Both transforms use complex variables; "s" and "z"
• Z-transform is for discrete-time systems and Laplace is for continuous time but their applications overlap

Relationship between Transforms

• Z transform can act as a discrete-time equivalent of the Laplace transform
• If a continuous-time signal is sampled, the Z-transform of the sampled signal can be related to the Laplace transform of the continuous-time signal

Transitioning Equations

• A continuous time signal is defined x(t)
• The Laplace transform is: X(s) = ∫₀^∞ x(t)e^(-st) dt
• Where s = σ + jω
• Sampling the continuous time input can derive the z transform from equation above

Sampled Singal Equations

• Equations for a sampled signal x(nTs), with Ts = 1:
• Thus x(nTs) → x(n)
• X(s) = ∫₀^∞ x(t)e^(-st) dt → X(e^s) = ∑(from n=0 to ∞) x(n)e^(-sn)
• Setting z = e^s leads to X(z) = ∑(from n=0 to ∞) x(n)z^(-n)
• The equation is for the Z transform

Similarities between Laplace, Fourier, and Z-Transforms

• Laplace equation defined as: X(s) = ∫₀^∞ x(t)e^(-st) dt
• Z Transform equation defined as: X(z) = ∑(∞, m=0) x(m)z^(-m)
• Fourier Transform defined as: X(z = e^(-j2πfm)) = ∑(∞, m=-∞) x(m)e^(-j2πfm)

Z Plane and Unit Circle

• The Laplace Transform is s = σ + jω
• The Z-Transform is z = re^(jθ)
• jω is the axis of the s-plane
• Inside unit circle corresponds to the σ < 0 part of the s-plane
• Outside unit circle corresponds to the σ > 0 part of the s-plane
• The unit circle, or the outer edge is the Fourier Transform in the z-plane

Bilateral Z-transform

• The bilateral or two-sided Z-transform of a discrete-time signal x[n] creates a power series X(z)
• Bilateral Z-transform as defined as: X(z) = Z{x[n]} = ∑(from n=-∞ to ∞) x[n]z^(-n)
• n is an integer and z is a complex number
• z = Ae^(jϕ) = A(cos ϕ + j sin ϕ), where:
• A is the magnitude of z, j is the imaginary unit, and ϕ is the complex argument (also defined as angle or phase)

Unilateral Z-Transform

• The single-sided or unilateral Z-transform is defined for instances where x[n] is only defined for n ≥ 0
• Expressed as X(z) = Z{x[n]} = ∑( ∞, n=0) x[n]z^(-n)
• This definition is useful when assessing a discrete-time causal systems unit impulse in signal processing

Z-Transform as an Operator

• The z-transform of a sequence x[n] is X(z) = ∑(∞, n=-∞) x[n]z^(-n)
• The z-transform acts as the operator Z{·} and transforms a sequence to a function.
• Z{x[n]} = ∑(∞, n=-∞) x[n]z^(-n) = X(z)
• In both cases z is a continuous complex variable

Z-Transforms and Fourier

• Fourier transform can obtained from the z-transform with substitution of z = e^(jω)
• Substitution of z = e^(jω) corresponds to restricting |z| = 1, with z = re^(jω)
• X(re^(jω)) = ∑(∞, n=-∞) xn^(-n) = ∑(∞, n=-∞) (x[n]r^(-n)) e^(-jωn)
• The z-transform becomes the Fourier transform of the sequence x[n]r^(-n)
• Fourier transform of x[n] is at r = 1
• The Fourier transform corresponds to the z-transform evaluated on the unit circle

Z plane

• On the z-plane: z = e^(jω)

Convergence

• Fourier transform does not converge for all sequences-- the infinite sum may not always be finite
• Similarly, the z-transform does not converge for all sequences or values of z
• The region of convergence (ROC) is the set of values of z for which z-transform converges

Transform of x[n]

• Fourier transform of x[n] exists if sum of ∑(∞, n=-∞)|x[n]| converges
• The z-transform of x[n] = Fourier transform of sequence x[n]r^(-n) if this converges
• X(z) = ∑(∞, n=-∞) |x[n]r^(-n)| < ∞
• Resulting condition: ∑(∞, n=-∞) |x[n]||z|^(-n) < ∞ meaning the ROC is a ring in the z-plane

Origin and Convergence

• In some instances, the inner radius includes origin with the outer radius extending to infinity
• Fourier transform converges if ROC contains the unit circle |z| = 1

Z-Transform Expression

• Most useful z-transforms are expressed in the form: X(z) = P(z)/Q(z)
• Where P(z) and Q(z) are polynomials in z
• P(z) = 0 are called the zeros of X(z), values with Q(z) = 0 are referred to as poles
• Zeros and poles fully define X(z) to within a multiplicative constant