Discrete transfer function in Simulink - Deep Dive

Overview - Discrete transfer function

What is it?

A discrete transfer function is a mathematical way to describe how a digital system changes input signals into output signals over time. It uses sequences of numbers to represent signals at specific time steps, unlike continuous functions that work with smooth signals. This function is often shown as a ratio of polynomials in a variable that represents a delay or shift in time. It helps us understand and design digital filters and controllers in systems like Simulink.

Why it matters

Without discrete transfer functions, designing and analyzing digital systems like controllers or filters would be much harder. They let engineers predict how a system reacts to inputs step-by-step, which is crucial for digital devices that work with sampled data. Without this concept, digital control and signal processing would lack a clear, simple way to model system behavior, making many modern technologies unreliable or impossible.

Where it fits

Before learning discrete transfer functions, you should understand basic signals and systems concepts, especially continuous transfer functions and sampling. After mastering discrete transfer functions, you can explore digital filter design, digital control systems, and advanced simulation techniques in Simulink.

Mental Model

Core Idea

A discrete transfer function describes how each output value depends on past inputs and outputs at specific time steps using a simple ratio of polynomials in a delay variable.

Think of it like...

Imagine a recipe that tells you how to bake a cake step-by-step, where each step depends on the results of previous steps and ingredients added earlier. The discrete transfer function is like that recipe for signals, showing how each output slice depends on past input slices.

Discrete Transfer Function Structure:

  Input sequence x[k] ---> [System] ---> Output sequence y[k]

  System defined by:
  Y(z)/X(z) = (b0 + b1 z^-1 + ... + bM z^-M) / (1 + a1 z^-1 + ... + aN z^-N)

  Where z^-1 represents a one-step delay.

  At each time k:
  y[k] = -a1 y[k-1] - ... - aN y[k-N] + b0 x[k] + b1 x[k-1] + ... + bM x[k-M]

Build-Up - 7 Steps

1

Foundation

Understanding discrete signals and sampling

Concept: Learn what discrete signals are and how continuous signals become discrete through sampling.

Signals in the real world are often continuous, like sound waves. To work with them digitally, we take snapshots at regular time intervals; this process is called sampling. The result is a sequence of numbers, each representing the signal value at a specific time step. These sequences are called discrete signals.

Result

You can represent any continuous signal as a sequence of numbers sampled at fixed intervals.

Understanding discrete signals is essential because discrete transfer functions operate on these sequences, not on continuous signals.

2

Foundation

Basics of transfer functions in continuous systems

Concept: Learn how continuous transfer functions describe system behavior using Laplace transforms.

In continuous systems, transfer functions relate input and output signals in the frequency domain using Laplace transforms. They are ratios of polynomials in the complex variable s, representing differentiation and integration. This helps predict how systems respond to inputs over time.

Result

You understand how continuous systems are modeled and analyzed using transfer functions.

Knowing continuous transfer functions sets the stage for understanding their discrete counterparts, which work similarly but with sequences.

3

Intermediate

Introducing the z-transform and delay operator

🤔Before reading on: do you think the z-transform converts sequences into continuous functions or into another sequence form? Commit to your answer.

Concept: The z-transform converts discrete sequences into a form that makes analysis easier, using a delay operator z^-1 to represent one time step delay.

The z-transform takes a sequence and turns it into a function of a complex variable z. The key idea is that z^-1 represents delaying the sequence by one time step. This lets us write equations for systems as ratios of polynomials in z^-1, simplifying analysis and design.

Result

You can represent discrete systems with algebraic equations in z, making them easier to study.

Understanding the z-transform and delay operator is crucial because discrete transfer functions are expressed as ratios of polynomials in z^-1.

4

Intermediate

Formulating discrete transfer functions

🤔Before reading on: do you think discrete transfer functions use sums of past inputs only, or both past inputs and outputs? Commit to your answer.

Concept: Discrete transfer functions relate current output to past outputs and inputs using polynomials in the delay operator.

A discrete transfer function is written as: H(z) = Y(z)/X(z) = (b0 + b1 z^-1 + ... + bM z^-M) / (1 + a1 z^-1 + ... + aN z^-N) This means the output at time k depends on past outputs and inputs weighted by coefficients a and b. This formula captures the system's behavior step-by-step.

Result

You can write and interpret discrete transfer functions that model digital systems.

Knowing that outputs depend on both past inputs and outputs helps understand feedback and memory in digital systems.

5

Intermediate

Implementing discrete transfer functions in Simulink

Concept: Learn how to use Simulink blocks to model discrete transfer functions for simulation.

Simulink provides a 'Discrete Transfer Function' block where you enter numerator and denominator coefficients. The block uses these to compute output sequences from input sequences during simulation. You can set the sample time to match your system's sampling rate.

Result

You can simulate digital systems using discrete transfer functions in Simulink.

Using Simulink blocks bridges theory and practice, letting you test and visualize system behavior easily.

6

Advanced

Stability and causality in discrete transfer functions

🤔Before reading on: do you think a discrete system is stable if all poles are inside or outside the unit circle in the z-plane? Commit to your answer.

Concept: Stability depends on the location of poles in the z-plane; causality means output depends only on current and past inputs.

For a discrete system to be stable, all poles (roots of the denominator polynomial) must lie inside the unit circle in the complex z-plane. This ensures outputs don't grow without bound. Causality means the system does not depend on future inputs, which is true if the transfer function is proper (degree denominator ≥ numerator).

Result

You can analyze and ensure your discrete system behaves well and realistically.

Understanding stability and causality prevents designing systems that behave unpredictably or unrealistically.

7

Expert

Effects of quantization and finite precision on discrete transfer functions

🤔Before reading on: do you think finite precision always preserves system stability? Commit to your answer.

Concept: Real digital systems use finite precision numbers, which can change system behavior and cause unexpected effects like instability or noise amplification.

When implementing discrete transfer functions on digital hardware, coefficients and signals are stored with limited bits. This quantization can shift pole locations, sometimes moving them outside the unit circle, causing instability. It can also introduce rounding noise. Engineers must carefully design and test systems considering these effects.

Result

You understand why simulations may differ from real hardware behavior and how to mitigate issues.

Knowing finite precision effects is vital for reliable real-world digital system design and avoiding subtle bugs.

Under the Hood

Discrete transfer functions work by representing system behavior as algebraic equations in the z-domain, where z^-1 acts as a delay operator. Internally, the system stores past input and output values in memory and computes the current output by multiplying these stored values by coefficients and summing them. This process is repeated at each time step, simulating the system's response to inputs over time.

Why designed this way?

This design leverages the z-transform to convert difference equations into algebraic forms, simplifying analysis and design. Using polynomials in z^-1 aligns naturally with digital systems' stepwise operation and memory of past values. Alternatives like time-domain difference equations are harder to analyze and optimize, so the transfer function approach became standard.

Discrete Transfer Function Internal Flow:

  +-------------------+
  | Past Inputs x[k-1], x[k-2], ... |
  +-------------------+
            |
            v
  +-------------------+      +-------------------+
  | Multiply by b coeffs | --> | Sum to form input part |
  +-------------------+      +-------------------+
            |
            v
  +-------------------+
  | Past Outputs y[k-1], y[k-2], ... |
  +-------------------+
            |
            v
  +-------------------+      +-------------------+
  | Multiply by a coeffs | --> | Sum to form feedback part |
  +-------------------+      +-------------------+
            |
            v
  +-------------------+
  | Combine sums and compute y[k] |
  +-------------------+

Myth Busters - 4 Common Misconceptions

Quick: Do you think discrete transfer functions only depend on past inputs, not past outputs? Commit to yes or no.

Common Belief: Discrete transfer functions only use past input values to compute the output.

Reality: They depend on both past inputs and past outputs, which allows feedback and memory in the system.

Why it matters: Ignoring past outputs misses feedback effects, leading to incorrect system models and unexpected behavior.

Quick: Do you think a discrete system is stable if poles lie outside the unit circle? Commit to yes or no.

Common Belief: A discrete system is stable as long as poles are not zero, regardless of their location.

Reality: Stability requires all poles to be inside the unit circle in the z-plane.

Why it matters: Misjudging stability can cause systems to produce outputs that grow without bound, causing failures.

Quick: Do you think the z-transform converts sequences into continuous time functions? Commit to yes or no.

Common Belief: The z-transform turns discrete sequences into continuous time signals.

Reality: The z-transform converts sequences into functions of a complex variable for analysis, not continuous time signals.

Why it matters: Confusing this leads to wrong interpretations and misuse of the transform in digital system design.

Quick: Do you think finite precision always preserves the designed system behavior? Commit to yes or no.

Common Belief: Finite precision in digital systems does not affect system stability or performance.

Reality: Finite precision can shift poles and zeros, causing instability or degraded performance.

Why it matters: Ignoring finite precision effects can cause real systems to fail even if simulations look correct.

Expert Zone

1

The choice of numerator and denominator polynomial orders affects system phase response and delay, which is critical in control applications.

2

Pole-zero cancellations in discrete transfer functions can hide unstable modes, leading to misleading stability analysis.

3

The sample time selection impacts the discrete transfer function's accuracy and stability, especially when converting from continuous models.

When NOT to use

Discrete transfer functions are not suitable for nonlinear or time-varying systems; in such cases, state-space models or time-domain simulations are better. Also, for very high-frequency signals or very fast dynamics, continuous-time models or hybrid approaches may be preferred.

Production Patterns

In real-world systems, discrete transfer functions are used in digital controllers like PID controllers implemented on microcontrollers, digital filters in audio processing, and communication systems. Engineers often combine them with state-space models and use software tools like Simulink for simulation and code generation.

Connections

Difference equations

Discrete transfer functions are algebraic representations of difference equations.

Understanding difference equations helps grasp how discrete transfer functions describe system dynamics step-by-step.

Digital filters

Discrete transfer functions define the behavior of digital filters used in signal processing.

Knowing discrete transfer functions allows designing filters that shape or clean signals in digital devices.

Economic time series models

Both use past values and inputs to predict current outputs over discrete time steps.

Recognizing this similarity helps apply system analysis techniques from engineering to economics and vice versa.

Common Pitfalls

#1 Confusing continuous and discrete transfer functions and applying continuous formulas directly to discrete systems.

Wrong approach:

Using Laplace variable s in place of z and ignoring sampling effects:
H(s) = (b0 + b1 s + ...)/(1 + a1 s + ...)

Correct approach:

Use z-transform variable and discrete polynomials:
H(z) = (b0 + b1 z^-1 + ...)/(1 + a1 z^-1 + ...)

Root cause:

Misunderstanding that discrete systems operate on sampled data and require different mathematical tools.

#2 Ignoring the sample time when modeling discrete transfer functions in Simulink.

Wrong approach:

Setting sample time to continuous or zero, causing simulation errors or incorrect results.

Correct approach:

Specify the correct fixed sample time matching the system's sampling rate in the block parameters.

Root cause:

Not recognizing that discrete systems must operate with defined time steps for accurate simulation.

#3 Assuming system stability without checking pole locations in the z-plane.

Wrong approach:

Designing a discrete transfer function without analyzing poles or assuming stability from continuous design.

Correct approach:

Calculate poles from denominator polynomial and verify they lie inside the unit circle.

Root cause:

Overlooking the fundamental stability criterion unique to discrete systems.

Key Takeaways

Discrete transfer functions model how digital systems transform input sequences into output sequences using polynomials in a delay variable.

They rely on the z-transform and delay operator to simplify analysis and design of digital filters and controllers.

Stability and causality depend on pole locations and polynomial degrees, which must be carefully checked.

Finite precision in real digital hardware can alter system behavior, so design must consider quantization effects.

Simulink provides practical tools to implement and simulate discrete transfer functions, bridging theory and application.

Practice

1. What does a discrete transfer function in Simulink represent?

easy

A. The color scheme of the Simulink blocks

B. The hardware configuration of the computer

C. The relationship between input and output in a step-by-step system

D. The file format used to save Simulink models

Solution

1. Step 1: Understand the purpose of discrete transfer functions

   A discrete transfer function models how the output of a system depends on its input at discrete time steps.

2. Step 2: Eliminate unrelated options

   Options about colors, hardware, or file formats do not describe system behavior.

3. Final Answer:

   The relationship between input and output in a step-by-step system -> Option C

4. Quick Check:

   Discrete transfer function = input-output relation [OK]

Hint: Think input-output step relation for discrete systems [OK]

Common Mistakes:

• Confusing transfer function with hardware settings
• Thinking it describes visual aspects
• Mixing file formats with system models

2. Which of the following is the correct way to define a discrete transfer function in Simulink?

easy

A. Using numerator and denominator coefficient vectors with a sample time

B. Using only a numerator vector without denominator

C. Using continuous time parameters without sample time

D. Using a single scalar value without vectors

Solution

1. Step 1: Recall discrete transfer function definition

   It requires numerator and denominator coefficient vectors plus a sample time to define the discrete system.

2. Step 2: Check options for completeness

   Only Using numerator and denominator coefficient vectors with a sample time includes numerator, denominator, and sample time, which is necessary.

3. Final Answer:

   Using numerator and denominator coefficient vectors with a sample time -> Option A

4. Quick Check:

   Numerator + denominator + sample time = correct syntax [OK]

Hint: Remember: numerator, denominator, and sample time needed [OK]

Common Mistakes:

• Omitting denominator coefficients
• Using continuous parameters for discrete blocks
• Using single scalar instead of vectors

3. Given a discrete transfer function with numerator = [0.5 0.5] and denominator = [1 -0.3], and sample time = 1, what is the output at step 1 if the input is 1 at step 0 and 0 afterwards?

medium

A. 0

B. 0.5

C. 0.65

D. 0.15

Solution

1. Step 1: Calculate output at step 1 using difference equation

   Output y[1] = 0.5*u[1] + 0.5*u[0] - (-0.3)*y[0]. Given u[0]=1, u[1]=0, y[0]=0 (initially), y[1] = 0.5*0 + 0.5*1 + 0.3*0 = 0.5.

2. Step 2: Confirm initial conditions and input values

   Initial output y[0] is zero, input at step 1 is zero, so calculation is correct.

3. Final Answer:

   0.5 -> Option B

4. Quick Check:

   y[1] = 0.5*0 + 0.5*1 + 0.3*0 = 0.5 [OK]

Hint: Use difference equation with initial zero output [OK]

Common Mistakes:

• Ignoring initial output conditions
• Using wrong input values for step 1
• Mixing numerator and denominator signs

4. You defined a discrete transfer function with numerator = [1 0] and denominator = [1 -0.5], sample time = 1. The output does not change when input changes. What is the likely error?

medium

A. Denominator coefficients are incorrect length

B. Sample time is set to zero

C. Numerator coefficients do not represent the input properly

D. Input signal is not connected

Solution

1. Step 1: Validate the coefficients

   Numerator [1 0] and denominator [1 -0.5] yield y[n] = u[n] + 0.5 y[n-1], which responds to input changes.

2. Step 2: Identify the likely cause

   In Simulink, if output ignores input despite valid parameters, the input signal is probably not connected to the block.

3. Final Answer:

   Input signal is not connected -> Option D

4. Quick Check:

   No response? Verify input connection [OK]

Hint: If output unresponsive, check input connection first [OK]

Common Mistakes:

• Ignoring numerator vector meaning
• Assuming sample time zero is valid
• Not verifying input connection

5. You want to design a discrete transfer function in Simulink to model a system with a delay of 2 steps and a gain of 0.8. Which numerator and denominator coefficients correctly represent this?

hard

A. Numerator = [0 0 0.8], Denominator = [1]

B. Numerator = [0.8 0 0], Denominator = [1 -0.2]

C. Numerator = [0 0.8 0], Denominator = [1 0]

D. Numerator = [0 0 1], Denominator = [0.8]

Solution

1. Step 1: Understand delay and gain in numerator

   A delay of 2 steps means the gain appears at the third coefficient (index 2), so numerator should be [0 0 0.8].

2. Step 2: Denominator for no feedback is [1]

   Since no feedback is mentioned, denominator is simply [1]. Other options have incorrect placements or values.

3. Final Answer:

   Numerator = [0 0 0.8], Denominator = [1] -> Option A

4. Quick Check:

   Delay = zeros before gain in numerator [OK]

Hint: Delay = zeros before gain in numerator vector [OK]

Common Mistakes:

• Placing gain at wrong numerator index
• Misunderstanding denominator role
• Confusing gain and delay positions