---
title: "Direct programming of a first-order filter"
subject: "Digital Control System"
chapter: 4
key: "semester_6/digital_control_system:direct-programming-first-order"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/direct-programming-first-order
site: BCT Engineering
---

# Direct programming of a first-order filter

Digital Control System · Chapter 4: Design and Compensation of Discrete Time Control System

- Asked in: 2071 Bhadra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/direct-programming-first-order
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-4

## Question

Construct the block diagram for the following pulse-transfer function system (a digital filter) by direct programming: G(z)=(2-0.6z⁻¹)/(1+0.5z⁻¹).

## Short answer

Direct programming uses the difference equation y(k)=-0.5y(k-1)+2x(k)-0.6x(k-1). The block diagram has feedforward gains 2 and -0.6, feedback gain -0.5, and two delay elements.

## Answer

#### Step 1: Difference equation
$$
\frac{Y(z)}{X(z)} = \frac{2 - 0.6z^{-1}}{1 + 0.5z^{-1}} \;\Rightarrow\; Y(z)(1 + 0.5z^{-1}) = X(z)(2 - 0.6z^{-1})
$$

$$
y(k) = -0.5y(k-1) + 2x(k) - 0.6x(k-1)
$$

#### Step 2: Direct programming block diagram
Each delayed input and output term is formed with its own delay element and added in a summer:

```text
x(k)
  o--[2]-------->(S)--------------o--> y(k)
  |               ^               |
[z^-1]            |             [z^-1]
  | x(k-1)        |               | y(k-1)
  o--[-0.6]----->(S)<[-0.5]-------o
```

- Feed-forward: $x(k)$ with gain 2, and $x(k-1)$ with gain $-0.6$.
- Feedback: $y(k-1)$ with gain $-0.5$.
- Direct programming uses $1 + 1 = 2$ delays.

For comparison, standard programming needs only one delay:
$w(k) = x(k) - 0.5w(k-1)$ and $y(k) = 2w(k) - 0.6w(k-1)$.

The pole at $z = -0.5$ is inside the unit circle, so the filter is stable.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. From G(z) = (2 - 0.6z^-1)/(1 + 0.5z^-1), the difference equation is:
   A. y(k) = 0.5y(k-1) + 2x(k) - 0.6x(k-1)
   B. y(k) = -0.5y(k-1) + 2x(k) - 0.6x(k-1)
   C. y(k) = -0.5y(k-1) + 2x(k) + 0.6x(k-1)
   D. y(k) = -0.5y(k-1) + 2x(k-1) - 0.6x(k)
   Answer: B. Cross-multiplying gives y(k) + 0.5y(k-1) = 2x(k) - 0.6x(k-1). (Key point: Difference equation)

2. How many delays does direct programming use here, compared with standard programming?
   A. 1 for direct, 2 for standard
   B. 2 for direct, 1 for standard
   C. 2 for both
   D. 1 for both
   Answer: B. Direct programming uses one delay for input and one for output (1+1 = 2); standard needs only 1. (Key point: Delays in direct programming)

3. The pole of the filter is at z = -0.5. What does this say about stability?
   A. Unstable, because it is negative
   B. Stable, because it is inside the unit circle
   C. Marginally stable
   D. Cannot be decided
   Answer: B. |z| = 0.5 < 1, so the pole is inside the unit circle and the filter is stable. (Key point: Stability)

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