How to Solve Logic Diagram to Output Latency Problems?

How to Solve Logic Diagram to Output Latency Problems?

In this type of problem, a gate-based circuit diagram or schematic is provided, and the task is to determine the output latency for a given set of gate delay functions.

What is Latency?

Latency in a combinational circuit is defined as the time interval between the initiation of an input signal and the appearance of the corresponding output signal. It reflects how long the circuit takes to respond to a change in input.

Ideal vs. Practical Assumptions

To simplify early-stage analysis, we often assume:

• Gates and wires are ideal, meaning they introduce no delay.
• This abstraction helps focus on logical behavior rather than timing intricacies.

However, in practical scenarios:

• Each logic gate introduces a propagation delay, which is the time between receiving a valid set of inputs and producing the corresponding output.
• Each wire also contributes to delay, influenced by factors such as length, cross-sectional area, and electromagnetic interference.

Why Propagation Delay Matters

Considering propagation delays is essential for:

• Determining the time at which the output becomes valid for a given input.
• Estimating the maximum operational frequency of the circuit, \(\mathsf{f=\frac{1}{T}}\), where \(\mathsf{T}\) is the latency.
• Identifying critical paths—the longest delay paths—which must be optimized to improve circuit speed.

Our Analytical Assumption

In our analysis:

• Our goal is to calculate the total latency from input to output, based on gate delays alone.
• We will assign propagation delays to gates.
• We will treat wires as ideal (zero delay).

Steps for Solution

A clear and structured set of steps for solving problems that ask you to calculate latency in digital logic circuits is given below:

Evaluating a Logic Circuit Example:

Now that we’ve outlined the steps for analyzing a logic circuit, let’s apply them to a specific example. To solve latency calculation problems in digital circuits, begin by identifying the input and output signals and tracing the direction of signal flow throughout the logic network. Then, for each distinct path leading to the output, compute the total delay by summing the propagation delays of all gates along that path. The path with the highest cumulative delay defines the overall latency of the circuit. Let’s assume the following delay for all the gates:

\[
\begin{align}
\textsf{NOT: }&\mathsf{10\,ps}\\
\textsf{NAND/NOR: }&\mathsf{20\,ps}\\
\textsf{AND/OR: }&\mathsf{30\,ps}\\
\textsf{XOR: }&\mathsf{40\,ps}\\
\textsf{XNOR: }&\mathsf{50\,ps}
\end{align}
\]

Step 1: Identify the input and output signals, and trace the direction of signal flow through the circuit.

Let’s begin by identifying the input and output signals of the circuit. In this example, the circuit includes three input signals—A, B, and C—and produces a single output signal, labeled Z. By convention, the diagram flows from left to right, which is typical in digital circuit design. However, keep in mind that some diagrams may flow from right to left, depending on how they’re presented.

Immediately after the input stage, you’ll notice that each input line branches off to a NOT gate, creating a parallel path. The output from each NOT gate is the complement of the respective input signal—\(\mathsf{A}\) becomes \(\mathsf{\overline{A}}\) or \(\mathsf{A’}\), \(\mathsf{B}\) becomes \(\mathsf{\overline{B}}\) or \(\mathsf{B’}\), and so on.

To aid readability, the inputs and their complements are color-coded, with \(\mathsf{A}\) and \(\mathsf{B}\) connected to the AND gate, and the NOT of signal \(\mathsf{B, \overline{B}}\) and \(\mathsf{C}\) connected to the OR gate.

This design technique is often used early in digital logic courses to help students recognize the relationship between an input and its complement. Later in the course, these input-complementing NOT gates are usually omitted visually—instead, the complemented form of the signal is directly represented at the gate input where needed. This saves space and removes unnecessary clutter. Circuit diagram without dedicated lines for the signal and its complement is shown here:

There are two paths to the output: one passes through the XOR and OR gates, the other through the NOT, NOR, and OR gates.

Step 2: For each path leading to the output, calculate the cumulative delay of all gates; the longest path determines the circuit’s latency.

With the inputs, output, and signal paths identified, we can now calculate the total propagation delay for each path. The path through the XOR and OR gates has delays of \(\mathsf{40ps}\) and \(\mathsf{30ps}\) respectively, totaling \(\mathsf{70ps}\). The alternate path through the NOT, NOR, and OR gates has delays of \(\mathsf{10ps, 20ps}\), and \(\mathsf{30ps}\), summing to \(\mathsf{60ps}\). Since the XOR–OR path has the greater delay, the circuit requires \(\mathsf{70ps}\) to produce a valid output when all inputs are applied simultaneously. Thus, the circuit’s latency is \(\mathsf{70ps}\). In other words, this circuit, on its own, can run at a frequency of around \(\mathsf{\left(\frac{1}{70\,ps}\right)\approx{}14.2\textsf{ GHz}}\).

This section focused on Logic Diagram to Latency, guiding you through its core concepts and practical examples. Whether you’re converting diagrams into Boolean expressions, tracing output values, or interpreting mathematical logic, each walk through is designed to build clarity and confidence. If you’re curious about other types of logic problems—like constructing circuits from mathematical notation, evaluating outputs using symbolic expressions, or calculating output latency across gate paths—you’ll find dedicated sections that break down each approach step by step. Dive into these categories to strengthen your understanding and navigate seamlessly between different problem-solving strategies for simple gate-based circuits.