Solution to MATLAB Assignment on Signals and Systems

Explore our comprehensive Sample Assignment Solution that provides a detailed solution to a sample MATLAB assignment focused on Signals and Systems. This sample assignment is designed to guide students through the fundamental concepts and practical applications of signal processing and system analysis using MATLAB.

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2. Welcome to the presentation of our sample assignment solution, designed to provide an insightful exploration into the fascinating topics of neural signals and system block diagrams. This assignment encompasses two main parts, offering a comprehensive understanding of complex neural and system processes. The first part delves into the measurement of action potentials in neurons. We examine how these electrical pulses travel along a neuron, specifically focusing on the scenario where the pulse moves in the positive z direction at a constant speed of 10 m/s. Exploring Neural Signals and System Block Diagrams: Sample Assignment Solution

3. The problem requires sketching the potential dependence on time at different positions along the neuron and determining the expression for the potential in terms of time and distance. The second part of the assignment involves characterizing a system defined by a block diagram. Here, the tasks include determining the system functional H, identifying the system's poles, and deriving the impulse response. These exercises are crucial for understanding how complex systems behave and respond to various inputs. This sample assignment solution is crafted to help students grasp the intricate concepts and methodologies in these areas, enhancing their problem-solving skills and theoretical knowledge.

4. The following figure illustrates the measurement of an action potential, which is an electrical pulse that travels along a neuron. Assume that this pulse travels in the positive z direction with constant speed ν = 10 m/s (which is a reasonable assumption for the large unmyelinated fibers found in the squid, where such potentials were first studied). Let Vm ( z,t ) represent the potential that is measured at position z and time t, where time is measured in milliseconds and distance is measured in millimeters. The right panel illustrates f(t) = Vm (30, t) which is the potential measured as a function of time t at position z = 30 mm. Problems 1. Neural signals

5. Sketch the dependence of Vm on t at position z = 40 mm (i.e., Vm (40, t)). Part a. It will take the action potention 1 ms to travel from the reference position at z = 30 mm to its new position at z = 40 mm. Thus, the new waveform Vm (40,t) is a version of f(t) that is shifted by 1 ms to the right. Solution:

6. Sketch the dependence of Vm on z at time t = 0 ms (i.e., Vm (z, 0)). Part b. Solution: The action potential peaks at z = 30 mm when t = 2 ms. Since it is traveling to the right at speed ν = 10 mm/ ms , it must also peak at z = 10 mm when t = 0. Thus f(2) must map to z = 10 mm in the new figure. Similarly, the following function locations map to new positions:

7. f(0) maps to 30 f(1) maps to 20 f(2) maps to 10 f(3) maps to 0 f(4) maps to −10

8. Determine an expression for Vm ( z,t ) in terms of f(·) and ν. Explain the relations between this expression and your results from parts a and b. Part c. Solution: Vm ( z,t ) =

9. The definition of f(t) provides a starting point: Vm (30,t) = f(t). In part a, we found that Vm (40,t) = f(t−1). This result generalizes: shifting to a more positive location (i.e., adding z0 to z) adds a time delay of z0/ν. Expressed as an equation, Vm (30 + z0,t) = f(t − z0 ν ). Substituting z = 30 + z0, we get the general relation Vm ( z,t ) =

10. To understand our result from part b, substitute t = 0 to obtain Vm (z, 0) = f(0 − z−30 ν ). Thus we must scale the x-axis by ν (to convert the time axis to a space axis) then shift the space axis by 30 mm (so that the peak is now at z = −10 mm) and finally, flip the plot about the x-axis (bringing the peak to z = 10 mm).

11. Problem 2. Characterizing block diagrams Consider the system defined by the following block diagram:

12. Determine the system functional H = Y/X . Part a. Let W represent the output of the topmost integrator. Then Solution: and

13. Substituting the former into the latter we find that Solving for Y/X yields the answer,

14. Determine the poles of the system. Part b. Substituting A → 1/s in the system functional yields Solution: The poles are then the roots of the denominator: −1 /2 , and −1.

15. Determine the impulse response of the system. Part c. Solution: Expand the system functional using partial fractions: Each term in the partial fraction expansion contributes one fundamental mode to h,

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