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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