8.4 Analysis of discrete-time systems using Z-transform

Discrete-time system modeling and analysis are crucial for understanding digital signal processing. These techniques use difference equations, Z-transforms, and transfer functions to represent and analyze systems, enabling engineers to design filters and predict system behavior.

Stability analysis in the Z-plane helps determine if a system's output remains bounded for bounded inputs. By examining pole locations and regions of convergence, engineers can ensure stable system performance and avoid unwanted oscillations or exponential growth in digital systems.

Discrete-Time System Modeling and Analysis

Modeling of discrete-time systems

Top images from around the web for Modeling of discrete-time systems

• 

  Z Transform How to Handle the Inital Condition - Electrical Engineering Stack Exchange View original

  Is this image relevant?

• 

  Computing the Z-transform in MATLAB | ee-diary View original

  Is this image relevant?

• 

  RC LPF Laplace Transform, Z-Transform & Bilinear Transform | ee-diary View original

  Is this image relevant?

• 

  Z Transform How to Handle the Inital Condition - Electrical Engineering Stack Exchange View original

  Is this image relevant?

• 

  Computing the Z-transform in MATLAB | ee-diary View original

  Is this image relevant?

1 of 3

Top images from around the web for Modeling of discrete-time systems

• 

  Z Transform How to Handle the Inital Condition - Electrical Engineering Stack Exchange View original

  Is this image relevant?

• 

  Computing the Z-transform in MATLAB | ee-diary View original

  Is this image relevant?

• 

  RC LPF Laplace Transform, Z-Transform & Bilinear Transform | ee-diary View original

  Is this image relevant?

• 

  Z Transform How to Handle the Inital Condition - Electrical Engineering Stack Exchange View original

  Is this image relevant?

• 

  Computing the Z-transform in MATLAB | ee-diary View original

  Is this image relevant?

1 of 3

• Difference equations establish the relationship between input and output sequences in the time domain
  • General form: $y[n] = \sum_{k=0}^{N} a_k y[n-k] + \sum_{k=0}^{M} b_k x[n-k]$
    • $y[n]$ represents the output sequence (audio signal)
    • $x[n]$ represents the input sequence (sensor data)
    • $a_k$ and $b_k$ are constant coefficients that define the system's behavior (filter parameters)
• Z-transform converts discrete-time sequences from the time domain to the Z-domain
  • Defined as: $X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$
  • Simplifies the analysis of discrete-time systems by transforming difference equations into algebraic equations (convolution becomes multiplication)
• Transfer functions in the Z-domain represent the input-output relationship of a discrete-time system
  • Obtained by taking the Z-transform of the difference equation and expressing the output in terms of the input
  • General form: $H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 - \sum_{k=1}^{N} a_k z^{-k}}$
  • Allows for the analysis of system properties such as stability and frequency response (low-pass filter)

Stability analysis in Z-plane

• Poles of a discrete-time system are the roots of the denominator polynomial of the transfer function
  • Determine the stability and behavior of the system based on their locations in the Z-plane
  • Stable systems have poles within the unit circle (|z| < 1), ensuring bounded output for bounded input (IIR filter)
• Stability conditions in the Z-plane categorize systems based on pole locations
  • Stable systems: all poles lie within the unit circle (|z| < 1)
  • Marginally stable systems: poles on the unit circle (|z| = 1) indicate oscillatory behavior (sinusoidal response)
  • Unstable systems: poles outside the unit circle (|z| > 1) lead to unbounded output (exponential growth)
• Region of convergence (ROC) is the set of values in the Z-plane for which the Z-transform converges
  • Determines the stability and causality of the system based on its relation to the poles
  • For a causal and stable system, the ROC includes the unit circle and extends outward (right-sided sequence)

Frequency Response and System Behavior

Frequency response using Z-transform

• Frequency response describes the system's response to sinusoidal inputs at different frequencies
  • Obtained by evaluating the transfer function on the unit circle: $H(e^{j\omega}) = H(z)|_{z=e^{j\omega}}$
    • $\omega$ is the normalized angular frequency ranging from 0 to 2π (Nyquist frequency)
• Magnitude response represents the gain of the system at different frequencies
  • Calculated as: $|H(e^{j\omega})|$
  • Determines the attenuation or amplification of frequency components (passband and stopband)
• Phase response represents the phase shift introduced by the system at different frequencies
  • Calculated as: $\angle H(e^{j\omega})$
  • Indicates the delay or advancement of frequency components (linear phase)

Behavior analysis with Z-transform techniques

• Transient response is the initial response of a system to an input
  • Determined by the poles and zeros of the system, which affect the decay rate and oscillations
  • Characterized by the speed of convergence to the steady-state (settling time)
• Steady-state response is the long-term behavior of a system
  • Determined by the input signal and the system's transfer function
  • For stable systems, the transient response decays, and the steady-state response dominates (step response)
• Inverse Z-transform techniques convert the Z-domain representation back to the time domain
  1. Partial fraction expansion decomposes the transfer function into simpler terms
     • Identify the residues and poles of the system
  2. Residue method calculates the residues of the partial fraction expanded terms
     • Determine the time-domain response using the residues and poles (impulse response)
  3. Long division divides the numerator by the denominator of the transfer function
     • Obtain the time-domain response from the coefficients of the resulting power series (convolution sum)