# Signals and Systems Jamshid Shanbehzadeh.

## Presentation on theme: "Signals and Systems Jamshid Shanbehzadeh."— Presentation transcript:

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

Outline 3 Sampling Nyquist Theorm Reconstruction Aliasing

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Unit Step

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Sifting Property

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Outline 1 Signals (Analog, Discrete, Digital) Analog Signals
Definition Special Analog signals Unit step Signum Sawtooth Dirac Delta Discrete Signals Polynomials and kindred signals Discrete Exponential signal Discrete Gaussian Signal Discrete Delta Sifting property Discrete Unit Step Conversion of analog to digital signals Absolutely summable signals Finite Energy Signals Cross Correlation Convolution Convolution Properties Associativity commutativity distributivity Linear systems Decomposition Translation invariant systems Linear Translation invariant systems Stable Systems

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

The Basics of Filtering in the Frequency Domain

A.i=3 A.j=2 A=3i+2j Y 2j x 3i

Inner Product A.B=(x1.y1+x2.y2+……xn.yn) f.g=?

f(t) = 2sin(wt) + 4sin(2wt)=2g(t)+4h(t)
f.g=2 f.h=4

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

Outline 2 Introduction Fourier Series Fourier Transform
Exponential Fourier Series Partial Fourier Series Examples Fourier Transform Definition Inverse Fourier Transform Fourier Transform Properties Time Shift Frequency Shift Scaling Time Differentiation Frequency Differentiation Fourier Transform of Unit step Complex exponential General periodic signal Sine and cosine Oscillations Convolution Theorem Discrete Fourier Transform (DFT) Definition Inverse of DFT DFT For Discrete periodic Signals DFT Properties Discrete Fourier Series (DFS) Inverse DFS Relation between DFS and FS Discrete Convolution Discrete-Time Fourier Transform (DTFT) Inverse DTFT Example (Discrete Delta, Discrete Pulse, Exponential Signal) DTFT Proprties (Linearity, Time Shift, Frequency Shift) Convolution Theorem

Discrete-Time Fourier Transform (DTFT)