1. COVOLUTION AND CORRELATION OF SIGNALS
UNIT IV
COVOLUTION AND CORRELATION OF SIGNALS

2. Convolution

3. Convolution Properties
Commutative:
f*g = g*f
Associative:
(f*g)*h = f*(g*h)
Homogeneous:
f*(g)=  f*g
Additive (Distributive):
f*(g+h)= f*g+f*h
Shift-Invariant
f*g(x-x0,y-yo)= (f*g) (x-x0,y-yo)

4. The Convolution Theorem
and similarly:

5. Examples
What is the Fourier Transform of ? *

6. Convolution

7. Definition of correlation
Correlational research determines to what degree a relationship exists between 2 variables (or more variables).

8. The nature of correlational research
Positive correlation means that high scores on one variable (X) tend to be associated with high scores on the other variable (Y).
Negative Correlation means that high scores on one variable (X) are associated with low scores on the other variable (Y).

9. Three Sets of Data Showing Different Directions and Degrees of Correlation  
X Y X Y X Y
(A) (B) (C)
r = r = r = 0

10. A negative correlationy x

11. No correlation y x

12. Purposes of Correlational Research
Correlational studies are carried out to explain important human behavior or to predict likely outcomes. (identify relationships among variables).
Explanatory studies
Prediction studies
More complex correlational techniques

13. Explanatory studies Prediction studies
To identify relationships among variables.
Prediction studies
If a relationship of sufficient magnitude exists between two variables, it becomes possible to predict score on one variable when score on related variable is known.
Predictor variable: The variable that is used to make the prediction.
Criterion variable: The variable about which the prediction is made.

14. Prediction Using a Scatterplot

15. More Complex Correlational Techniques
Multiple Regression
Coefficient of multiple correlation(R)
Coefficient of Determination
Discriminant Function Analysis
Factor Analysis
Path Analysis
Structural Modeling

16. More Complex Correlational Techniques
Multiple Regression
Technique that enables researchers to determine a correlation between a criterion variable and the best combination of two or more predictor variables.
Coefficient of multiple correlation(R)
Indicates the strength of the correlation between the combination of the predictor variables and the criterion variable

17. More Complex Correlational Techniques
Coefficient of Determination
Indicates the percentage of the variability among the criterion scores that can be attributed to differences in the scores on the predictor variable.
Discriminant Function Analysis
Rather than using multiple regression, this technique is used when the criterion value is categorical.

18. More Complex Correlational Techniques
Factor Analysis
Allows the researcher to determine whether many variables can be described by a few factors.
Path Analysis
Used to test the likelihood of a causal connection among three or more variables.
Structural Modeling
Sophisticated method for exploring and possibly confirming causation among several variables.

19. Path Analysis Diagram

20. Correlation coefficient
A decimal number between .00 and or –1.00 that indicates the degree to which two quantitative variables are related.
-1.00
0.00
+1.00
strong negative
strong positive
no relationship

21. PSD Consider a signal x(t) with Fourier Transform (FT) X(w)
We wish to find the energy and power distribution of x(t) as a function of frequency

22. Power and Energy Spectral Density
The power spectral density (PSD) Sx(w) for a signal is a measure of its power distribution as a function of frequency
It is a useful concept which allows us to determine the bandwidth required of a transmission system
We will now present some basic results which will be employed later on