Signals and Systems

9.8 System Function Algebra & Block Diagrams

Imron Rosyadi

Introduction: System Function Algebra

The Laplace Transform simplifies LTI system analysis by converting time-domain operations into algebraic ones. This is particularly useful for:

• Analyzing interconnections of LTI systems.
• Synthesizing systems from elementary building blocks.

Tip

Key Idea:

Convolution in the time domain becomes multiplication in the s-domain. Differentiation becomes multiplication by \(s\). Integration becomes multiplication by \(1/s\).

Parallel Interconnection

Consider two LTI systems, \(H_1(s)\) and \(H_2(s)\), connected in parallel.

Time Domain:

The impulse response of the overall system is the sum of individual impulse responses:

\[h(t) = h_1(t) + h_2(t)\]

S-Domain:

Due to the linearity of the Laplace transform, the overall system function is the sum of individual system functions:

\[H(s) = H_1(s) + H_2(s)\]

Parallel Interconnection

Figure 9.30(a): Parallel interconnection of two LTI systems.

In a parallel interconnection, the input signal \(x(t)\) is applied to both systems simultaneously. The outputs of the individual systems, \(y_1(t)\) and \(y_2(t)\), are then summed to produce the overall output \(y(t)\). This is analogous to connecting two resistors in parallel, where the total conductance is the sum of individual conductances. This setup is common in audio mixing or parallel processing.

Interactive Example: Parallel Systems

Let’s explore the effect of parallel interconnection on system response. Consider two first-order systems: \(H_1(s) = \frac{1}{s+a}\) and \(H_2(s) = \frac{1}{s+b}\).

viewof pole_a = Inputs.range([0.1, 5.0], {value: 1.0, step: 0.1, label: "Pole 1 (a)"});
viewof pole_b = Inputs.range([0.1, 5.0], {value: 2.0, step: 0.1, label: "Pole 2 (b)"});

Adjust the pole values ‘a’ and ‘b’ using the sliders. Observe how the individual step responses of \(H_1(s)\) and \(H_2(s)\) combine to form the step response of the overall parallel system. Notice that the parallel system’s response is simply the sum of the individual responses, demonstrating the linearity of the system function algebra. This interactive plot helps visualize the direct relationship between individual system characteristics and the combined system’s behavior.

Series Interconnection

Consider two LTI systems, \(H_1(s)\) and \(H_2(s)\), connected in series (cascade).

Time Domain:

The impulse response of the overall system is the convolution of individual impulse responses:

\[h(t) = h_1(t) * h_2(t)\]

S-Domain:

The overall system function is the product of individual system functions:

\[H(s) = H_1(s) H_2(s)\]

Series Interconnection

Figure 9.30(b): Series combination of two LTI systems.

In a series interconnection, the output of the first system, \(y_1(t)\), becomes the input to the second system, \(x_2(t)\). The final output \(y(t)\) is then the output of the second system. This setup is fundamental in signal processing chains, where multiple filters or processing stages are applied sequentially. For example, an audio signal might pass through an equalizer (filter 1) and then a compressor (filter 2).

Interactive Example: Series Systems

Let’s see how cascading two first-order systems affects the overall step response.

Systems: \(H_1(s) = \frac{1}{s+a}\) and \(H_2(s) = \frac{1}{s+b}\).

viewof pole_c = Inputs.range([0.1, 5.0], {value: 1.0, step: 0.1, label: "Pole 1 (c)"});
viewof pole_d = Inputs.range([0.1, 5.0], {value: 2.0, step: 0.1, label: "Pole 2 (d)"});

Similar to the parallel case, adjust the pole values ‘c’ and ‘d’. Notice how the series connection results in a system that is often “slower” or more filtered than the individual components, especially if the poles are close to the origin. The overall pole locations are determined by the product of the individual system functions. This is crucial for designing multi-stage filters or control systems where each stage contributes to the overall dynamics.

Feedback Interconnection

Feedback is a powerful concept in engineering, fundamental to control systems, amplifiers, and oscillators.

System Equations:

From the diagram:

\(Y(s) = H_1(s) E(s)\)

\(E(s) = X(s) - Z(s)\)

\(Z(s) = H_2(s) Y(s)\)

Substituting and solving for \(Y(s)/X(s)\):

\(Y(s) = H_1(s) [X(s) - H_2(s) Y(s)]\)

\(Y(s) = H_1(s) X(s) - H_1(s) H_2(s) Y(s)\)

\(Y(s) [1 + H_1(s) H_2(s)] = H_1(s) X(s)\)

Overall System Function: \[H(s) = \frac{Y(s)}{X(s)} = \frac{H_1(s)}{1 + H_1(s) H_2(s)}\]

Feedback Interconnection

Figure 9.31: Feedback interconnection of two LTI systems.

Feedback systems are ubiquitous in engineering. A simple example is a thermostat controlling room temperature. The sensor (\(H_2(s)\)) measures the actual temperature (\(Y(s)\)), compares it to the desired temperature (\(X(s)\)), and sends an error signal (\(E(s)\)) to the heating/cooling system (\(H_1(s)\)). The primary goal of feedback is to regulate or stabilize a system, improve its performance, or make it less sensitive to disturbances. The algebraic derivation shows how feedback fundamentally alters the system’s poles, which directly impacts stability and response characteristics.

Block Diagram Representations

Block diagrams are visual tools to represent LTI systems, especially those described by differential equations. They use three basic operations:

1. Addition: Summing junction.
2. Multiplication by a coefficient: Gain block.
3. Integration: Integrator block (\(1/s\)).

Important

These building blocks allow us to visualize the internal structure and signal flow within a system.

Example 9.28: First-Order System

Consider a causal LTI system with system function \(H(s)=\frac{1}{s+3}\).

This corresponds to the differential equation: \(\frac{d y(t)}{d t}+3 y(t)=x(t)\).

We can rearrange the differential equation as:

\(\frac{d y(t)}{d t} = x(t) - 3 y(t)\)

Integrating both sides:

\(y(t) = \int [x(t) - 3 y(t)] dt\)

This suggests a feedback structure where the input to the integrator is \(x(t) - 3y(t)\).

Example 9.28: First-Order System

Let’s visualize this using a block diagram.

Figure 9.32(a): Block diagram representation.

This example demonstrates how a simple first-order differential equation can be translated into a block diagram using an integrator, a gain block, and a summing junction. The term \(1/s\) represents the integrator, which performs the operation of integration in the s-domain. The feedback path with gain -3 represents the term \(3y(t)\) being subtracted from the input. This direct translation from a differential equation to a block diagram is a fundamental skill in system analysis and design.

Example 9.29: First-Order System with Zero

Consider \(H(s)=\frac{s+2}{s+3}\).

This can be written as \(H(s)=\left(\frac{1}{s+3}\right)(s+2)\), suggesting a cascade of two systems.

The first system is \(H_A(s) = \frac{1}{s+3}\), which we just saw.

The second system is \(H_B(s) = s+2\).

If \(Z(s)\) is the output of \(H_A(s)\), then \(Y(s) = (s+2)Z(s)\).

In time domain: \(y(t) = \frac{d z(t)}{d t} + 2 z(t)\).

The input to the integrator in the \(H_A(s)\) block is \(e(t) = \frac{d z(t)}{d t}\).

So, \(y(t) = e(t) + 2 z(t)\).

This avoids explicit differentiation, leading to a “direct form” representation.

Example 9.29: First-Order System with Zero

Figure 9.33(b): Equivalent block diagram representation.

This example shows how a system with both poles and zeros can be represented. The “tapping” of signals from within the first stage (specifically the input to the integrator, \(e(t)\), which is \(dz(t)/dt\)) is a common technique to avoid explicit differentiators, which are often problematic in practical implementations due to noise amplification. This direct-form structure is efficient and widely used in digital filter implementations.

Example 9.30: Second-Order System

Consider \(H(s)=\frac{1}{(s+1)(s+2)}=\frac{1}{s^{2}+3 s+2}\).

This corresponds to the differential equation: \(\frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=x(t)\).

Direct Form:

Rearranging the differential equation:

\(\frac{d^{2} y(t)}{d t^{2}} = x(t) - 3 \frac{d y(t)}{d t} - 2 y(t)\)

We can use two integrators in cascade to obtain \(y(t)\) from \(\frac{d^{2} y(t)}{d t^{2}}\).

The outputs of these integrators provide \(\frac{d y(t)}{d t}\) and \(y(t)\).

Example 9.30: Second-Order System

Figure 9.34(a): Direct form representation.

The direct form for a second-order system extends the ideas from the first-order case. Two cascaded integrators are used, where the output of the first integrator is the first derivative of the output, and the output of the second is the output itself. Feedback paths from these outputs, scaled by the coefficients of the differential equation, are summed with the input to provide the second derivative, closing the loop. This structure is very straightforward to derive from the differential equation or system function.

Example 9.30: Cascade and Parallel Forms

For \(H(s)=\frac{1}{(s+1)(s+2)}\), other representations exist:

Cascade Form:

\(H(s)=\left(\frac{1}{s+1}\right)\left(\frac{1}{s+2}\right)\)

This is a series connection of two first-order systems.

Figure 9.34(b): Cascade form.

Example 9.30: Cascade and Parallel Forms

Parallel Form:

Using partial-fraction expansion:

\(H(s)=\frac{1}{s+1}-\frac{1}{s+2}\)

This is a parallel connection of two first-order systems.

Figure 9.34(c): Parallel form.

The cascade and parallel forms offer alternative ways to implement the same system. The cascade form breaks down the system into a sequence of simpler, often first-order, systems. This can be advantageous for modular design and analysis. The parallel form decomposes the system into a sum of simpler systems, typically through partial fraction expansion. This form can be useful when different modes of the system (corresponding to the poles) need to be processed independently, or for stability analysis. All three forms (direct, cascade, parallel) are equivalent in terms of their input-output behavior for LTI systems, but they differ in internal structure, which can impact sensitivity to component variations, numerical stability in digital implementations, and computational complexity.

Interactive Example: Second-Order System Response

Let’s observe the step response of a second-order system and how its poles affect it.

Consider \(H(s) = \frac{1}{s^2 + (p_1+p_2)s + p_1 p_2}\).

viewof pole_p1 = Inputs.range([0.1, 5.0], {value: 1.0, step: 0.1, label: "Pole 1 (p1)"});
viewof pole_p2 = Inputs.range([0.1, 5.0], {value: 2.0, step: 0.1, label: "Pole 2 (p2)"});

Adjust the pole locations \(p_1\) and \(p_2\) using the sliders. Observe how changing these values affects the transient response characteristics such as rise time, settling time, and overshoot (though this system is overdamped, so no overshoot here). For instance, moving poles closer to the origin (smaller positive values) generally makes the system slower, while moving them further away makes it faster. This interactive tool helps build an intuitive understanding of the relationship between pole locations and time-domain behavior, which is critical in control system design.

Example 9.31: More Complex Second-Order System

Consider \(H(s)=\frac{2 s^{2}+4 s-6}{s^{2}+3 s+2}\).

This system includes zeros in addition to poles.

Direct Form:

Similar to Example 9.29, we can tap signals from the integrators to realize the numerator polynomial.

This method allows the coefficients of the system function to directly appear in the block diagram.

Example 9.31: More Complex Second-Order System

Figure 9.35: Direct-form representation.

This direct-form representation is very general and can be applied to any rational system function. The denominator coefficients determine the feedback path (poles), and the numerator coefficients determine the feed-forward path (zeros). This structure is widely used in digital signal processing for implementing IIR (Infinite Impulse Response) filters due to its computational efficiency. Understanding how to derive and interpret these block diagrams is crucial for both analyzing existing systems and synthesizing new ones.

Conclusion: Power of System Function Algebra

• Algebraic Simplification: Laplace transform converts complex time-domain operations (convolution, differentiation) into simpler algebraic manipulations.
• System Interconnections: Provides straightforward methods to find the overall system function for parallel, series, and feedback configurations.
• Block Diagram Synthesis: Enables visual representation of LTI systems using basic building blocks (integrators, gains, summers).
• Multiple Implementations: A single system function can have various block diagram forms (direct, cascade, parallel), each with practical implications for design and implementation.

Note

Mastering system function algebra and block diagrams is fundamental for analyzing, designing, and implementing LTI systems in various ECE applications, from control systems to signal processing and communications.