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Signals and Systems Jamshid Shanbehzadeh
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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
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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
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Outline 3 Sampling Nyquist Theorm Reconstruction Aliasing
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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
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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
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Unit Step
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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
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Sifting Property
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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
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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
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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
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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
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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
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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
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The Basics of Filtering in the Frequency Domain
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A.i=3 A.j=2 A=3i+2j Y 2j x 3i
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Inner Product A.B=(x1.y1+x2.y2+……xn.yn) f.g=?
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f(t) = 2sin(wt) + 4sin(2wt)=2g(t)+4h(t)
f.g=2 f.h=4
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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
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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
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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
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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
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Discrete-Time Fourier Transform (DTFT)
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