EE 438 Essential Definitions and Relations
ECE 438 Essential Definitions and Relations
CT Metrics
1. Energy [pic]
2. Power[pic]
3. root mean squared value [pic]
4. Area [pic]
5. Average value [pic]
6. Magnitude [pic]
DT Metrics
1. Energy [pic]
2. Power [pic]
3. root mean squared value [pic]
4. Area [pic]
5. Average [pic]
6. Magnitude [pic]
CT Signals and Operators
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
Properties of the CT Impulse
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
CTFT
Sufficient conditions for existence
[pic]
Forward transform
[pic]
Inverse transform
[pic]
Transform Relations
1. Linearity
[pic]
2. Scaling and shifting
[pic]
3. Modulation
[pic]
4. Reciprocity
[pic]
5. Parseval
[pic]
6. Initial Value
[pic]
7. Convolution
[pic]
8. Product
[pic]
9. Transform of periodic signal
[pic]
10. Transform of sampled signal
[pic]
Transform Pairs
1. [pic]
2. [pic]
3. [pic]
4. [pic]
DT Signals and Operators
1. [pic]
2. [pic]
3. [pic]
4. Downsampling [pic]
[pic]
5. Upsampling [pic]
[pic]
Properties of the DT Impulse
1. [pic]
2. [pic]
3. [pic]
DTFT
Sufficient conditions for existence
[pic]
Forward transform
[pic]
Inverse transform
[pic]
Transform Relations
1. Linearity
[pic]
2. Shifting
[pic]
3. Modulation
[pic]
4. Parseval
[pic]
5. Initial Value
[pic]
6. Convolution
[pic]
7. Product
[pic]
8. Downsampling [pic]
[pic]
9. Upsampling [pic]
[pic]
Transform Pairs
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
Relation between CT and DT
1. Time Domain
[pic]
2. Frequency Domain
[pic]
ZT
Forward transform
[pic]
Transform Relations
1. Linearity
[pic]
2. Shifting
[pic]
3. Modulation
[pic]
4. Multiplication by time index
[pic]
5. Convolution
[pic]
6. Relation to DTFT
If [pic] converges for [pic], then the DTFT of [pic] exists and
[pic]
Transform Pairs
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
DFT
Forward transform
[pic]
Inverse transform
[pic]
Transform Relations
1. Linearity
[pic]
2. Shifting
[pic]
3. Modulation
[pic]
4. Reciprocity
[pic]
5. Parseval
[pic]
6. Initial Value
[pic]
7. Periodicity
[pic]
8. Relation to DTFT of a finite length sequence
Let [pic] only for [pic]
[pic].
Define [pic],
[pic],
then [pic].
9. Periodic convolution
[pic]
10. Product
[pic]
Transform Pairs
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
Random Signals
1. Probability density function [pic]
[pic]
[pic]
2. Probability distribution function [pic]
[pic]
[pic]
[pic] is a non-decreasing function of [pic]
Expectation Operator
[pic]
1. Linearity of expectation
[pic]
2. Mean [pic]
3. 2nd moment [pic]
4. Variance [pic]
5. Relation between mean, 2nd moment, and variance
[pic]
Miscellaneous
1. Geometric Series
[pic]
2. [pic] roots [pic], of a complex constant [pic], i.e. [pic] solutions to
[pic]
[pic]
3. Trigonometric Identities
[pic]
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