Chapter 2. Math Tools. Prince.

Question 1

Point impulse (impulse function)

Answer 1

Is used as the input signal to characterize the response of a system, which is the impulse response.

Question 2

Comb and sampling functions, rect and sinc functions, and exponential and sinusoid all signals

Answer 2

Are used to help describe and characterize signaling systems

Question 3

A system is linear if

Answer 3

When the input consists of a collection of signals, the output is a summation of the responses of the system to each individual input signal

Question 4

A system is shift invariant if

Answer 4

An arbitrary translation of the input signal results in an identical translation of the output

Question 5

A signal is separable if

Answer 5

It can be represented as the product of 1D signals

Question 6

Fourier transform

Answer 6

Represents signals as a sum of sinusoids of different frequencies with associated magnitude and phase

Question 7

Transfer function

Answer 7

The Fourier space analog of the impulse response

Question 8

Output of a linear shift invariant system

Answer 8

The convolution of the input with the impulse response; in Fourier space, it is the product of the Fourier transform of the input and the transfer function

Question 9

Continuous signals

Answer 9

Are transformed by sampling or discretization into discrete signals in order to be digitally represented

Question 10

Aliasing

Answer 10

Is caused by improper sampling of a continuous signal, yielding artifacts in the resulting digital image

Question 11

Signals

Answer 11

Mathematical functions. They can be discrete, continuous or mixed.

Question 12

A system is linear if it satisfies:

Answer 12

Question 13

Sifting property of the delta function

Answer 13

Question 14

Linear property of delta function

Answer 14

Question 15

L has the property of “shift invariance”

Answer 15

Question 16

Linear Shift Invariant system (LSI)

Answer 16

The output is determined by the convolution of the input and the PSF

Question 17

Convolution

Answer 17

Question 18

Connections of LSI Systems:

Answer 18

Question 19

Seperable systems

Answer 19

Question 20

1D Fourier Transform

Answer 20

Question 21

Rect and Sinc

Answer 21

Question 22

Fourier transform relationship of rect and sinc

Answer 22

Question 23

Linearity of Fourier Transform

Answer 23

Question 24

Scaling of Fourier Transform

Answer 24

Question 25

Shifting Property of Fourier transform

Answer 25

Question 26

Convolution Property of Fourier Transform

Answer 26

Question 27

Separable input property of Fourier Transform

Answer 27

Question 28

Parseval’s Theorem Fourier Transform

Answer 28

Question 29

Product Property of Fourier transform

Answer 29

Question 30

Impulse Property of Fourier transform

Answer 30

Question 31

Constant property of Fourier Transform

Answer 31

Question 32

comb function

Answer 32

Question 33

Fourier Transform of the comb function

Answer 33

Question 34

sampling function

Answer 34

Question 35

scaling property of the delta function

Answer 35

Question 36

relation of sampling function and comb function

Answer 36

Question 37

Sampled signal is

Answer 37

Question 38

FT of the sampled signal

Answer 38

Question 39

Fourier Transform

Answer 39

Question 40

Convolution of Fourier Transform

Answer 40

Question 41

Parseval’s Theorem

Answer 41

Question 42

Product of Fourier Transform

Answer 42