
- Deeper academic knowledge, especially in advanced math/science topics
- Enhanced critical thinking and problem-solving abilities
- Improved study and test-taking strategies
- Higher levels of self-motivation and dedication
Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.
Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.
Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.
Sampling
The sampling theorem states that in order to accurately reconstruct a continuous-time signal from its samples, the sampling rate must be at least twice the highest frequency component in the signal. This minimum sampling rate is known as the Nyquist rate. If the sampling rate is below the Nyquist rate, aliasing occurs, leading to distorted or misleading reconstructions.
Based on the FE Electrical and Computer topic list, the subtopics covered in the Digital Filters are Difference Equation and Z-Transform. However, Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) shall not be neglected even though they were not listed.
1. Difference EquationA general form of a difference equation is:
y[n]=a0 x[n]+a1 x[n-1]+...+b0 y[n-1]+b1 y[n-2]+... (x)
Where,
y[n] is the current output at discrete time n.
x[n] is the current input at discrete time n.
x[n-1],... is the past input values.
y[n-1],... is the past output values.
a0, … and b0,... are the coefficients of the difference equation.
Example:
y[n]=4x[n]+2y[n-1]+2y[n-2]; y[n]=0 for n<0
What is the impulse response of a system described by the difference equation?
Solution:
1. For impulse response, x[n]=δ[n] and δ[0]=1 and δ[n≥1]=0
. n=0
y[0]=4δ[0]+2y[0-1]+2y[0-2]=4(1)+y[-1]+y[-2]=4+0+0=4
. n=1
y[1]=4δ[1]+2y[1-1]+2y[1-2]=4(0)+2y[0]+2y[-1]=0+2(4)+2(0)=8
. n=2
y[2]=4δ[2]+2y[2-1]+2y[2-2]=4(0)+2y[1]+2y[0]=4+2(8)+2(4)=28
You can keep computing for n approaches ∞ but the solution right is h[n]={4.8,28,...}
2. Z-Transform∞ | |
X(z)=∑ | x[n]z-n(x) |
n = 0 |
In the exam, a table of z transform paris is provided in the Reference Handbook p.372.
Example:
Find the z-transform X(z) of the following signal:
x[n]=δ[n]+1/5 δ[n-2]-3/4 δ[n-4]
Solution:
1. Use the summation definition (equation x) to solve for the z transform:
∞ | |
X(z)=∑ | x[n]z-n |
n = 0 |
2. x[n] is zero for all values of n except when n=0,2,and 4.
x[n]={1,0,1/5,0,-3/4}
. n=0, z transform is
x[0]z-0=1⋅1=1
. n=2, z transform is
x[2]z-2=1/5⋅z-2
. n=4, z transform is
x[4]z-2=-3/4⋅z-4
So, X(z)=1+1/5 z-2-3/4 z-4
3. Finite Impulse Response (FIR)The output of the FIR filter can be calculated as:
∞ | |
Y(n)=∑ | bnx[n-k] |
k = 0 |
The output of the IIR filter can be calculated as:
y[n] = ∑l=0∞ bl x[n-l] + ∑k=0∞ ak y[n-k]
Example:
With the given filter below
y[n]=x[n]+1/6 x[n-1]
What type of filter is it?
Solution:
1. By looking at the function, the output contains the sum of the current and past inputs. Therefore, it is an FIR filter.
2. Do a z-transform to determine if it is a low-pass, high-pass, band-pass, or band-reject filter.
y[n]=x[n]-1/6 x[n-1]
Use the z-transform pair from Reference Handbook p.372, we have
Y(z)=X(z)-1/6 X(z)z^(-1)
H(z)=(Y(z))/(X(z))=1-1/6 z^(-1)=(z-1/6)/z
This blog aims to serve as a valuable point of reference, offering concise explanations and practical examples for each topic addressed in the exam. It is important to note, however, that while the blog provides a helpful overview, it may not encompass all aspects related to digital signal processing. For a more thorough and comprehensive study plan, I recommend visiting School of PE’s FE Electrical exam review course FE Electrical exam review course, which offers an extensive and detailed curriculum to ensure a comprehensive understanding of the subject matter.
References
D, P., GJ, A., & D, F. (2001). Neuroscience 2nd edition. Sunderland (MA): Sinauer Associates. https://www.ncbi.nlm.nih.gov/books/NBK10924/
Hasegawa, M. (2021). Lecture 6: Sampling and Aliasing. Retrieved June 6, 2023, from https://courses.engr.illinois.edu/ece401/fa2021/lectures/lec06.pdf
McClellan, J. H., Schafer, R. W., & Yoder, M. A. (2021). DSP First Second Edition. Pearson.
Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.
Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.