Lab 9: Convolution TIMS Week 2

 

Introduction 

Lab 9 serves as a second part to the previous TIMS convolution lab, which should be referenced to effectively understand the significance of this lab. Convolution is a mathematical operation used to express the relationship between three functions: an input signal, an output signal, and an impulse response. It is a fundamental tool in signals and systems because it characterizes how a system transforms an input signal into an output signal. In essence, it integrates the effect of an input signal's past values on the present output, which is crucial for understanding and designing systems in various fields like electrical engineering, control systems, and digital signal processing. Convolution is important because it helps predict how systems respond to different inputs, which is essential for analyzing and designing filters, audio signals, communication systems, and more. 

 

  

Procedure 

Part I 

At the beginning of the lab, we recreated and confirmed the signals from the first week before proceeding. The output depicted in the figures can be interpreted as the result of three distinct signals that have been filtered and then combined into a final output.  

 

This process is analogous to the behavior of an inductor, as the signals exhibit a specific time delay, similar to how inductors, which are passive elements, store and filter energy. Therefore an inductor component creates the presence of ‘delayed energy’. 

 

Following with a definition of superposition: In signals and systems, the principle of superposition is a fundamental property that applies to linear systems. It states that the response of a linear system to a combination of inputs is the sum of the responses to each individual input. In other words, if a system is linear, then the output signal produced by two or more inputs applied together is the same as the sum of the outputs that would have been produced by each input individually. This principle is crucial because it allows complex signals to be broken down into simpler components, analyzed separately, and then summed to understand the system's overall behavior.  

 

From earlier observations, it can be concluded that if four input pulses occur in succession, their combined effect would produce an output that is the convolution of these four signals. Essentially, the output would manifest as the cumulative sum of these four inputs. 

 

In the present TIMS arrangement, one could realize this by creating new systems that capitalize on the SYNC output from the Sequence Generator, with all four signals undergoing filtration into the final output. This would visually demonstrate the system's convolution using physical hardware. 

 

Completing the handout requires certain steps to be fulfilled: 

 

 

Figure 1: Convolution handout 

 

In this activity, we fed a rectified sinusoidal wave into the Sample and Hold (S/H) section of the Z-1 module, which captures and temporarily retains the signal, thus converting it to a discrete form. Inputting the predefined gains b0, b1, and b2 into the SFP software simplified the decomposition of the output signal into a series of delayed inputs. This allowed a set of discrete values to traverse a system with a known impulse response, culminating in an output that is the sum of these individual contributions. This process is depicted in Figure 1, which illustrates the input, output, and the gains b0, b1, and b2. The figure demonstrates how at each time increment, the output is a cumulative result of the three staggered and time-shifted inputs.

 

Equation 1: At each individual time interval, the output is the cumulative result of the three inputs that have been delayed and shifted. 

 

Equations for y[1], y[2], and y[6] are as follows: 

𝑦[1] = ℎ[0] ∙ 𝑥[1] + ℎ[1] ∙ 𝑥[0] + ℎ[0] ∙ 𝑥[1] …  (2) 

𝑦[2] = ℎ[0] ∙ 𝑥[2] + ℎ[1] ∙ 𝑥[2] + ℎ[2] ∙ 𝑥[0] …  (3) 

𝑦[6] = ℎ[0] ∙ 𝑥[6] + ℎ[1] ∙ 𝑥[5] + ℎ[2] ∙ 𝑥[4] … (4) 

 

  

Conclusion 

The following lab was an interesting change from normal operation of procedures as it was mainly up to the student to figure out the solutions, but this proved to be difficult. For future reference I would include more clear instructions to not confuse the lab user. Superposition and convolution are crucial in signals and systems because they allow for the analysis and understanding of complex systems by breaking them down into simpler components. Superposition, which applies only to linear systems, enables the prediction of a system's response to multiple simultaneous inputs by adding up the individual responses. Convolution, on the other hand, provides a method to combine an input signal with a system's impulse response to find the output signal. Both principles are fundamental for designing and analyzing systems in telecommunications, signal processing, control systems, and electronics, as they help in understanding how signals interact with and propagate through systems.