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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– Definition– 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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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– Definition– 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– Definition– 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– Definition– 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– Definition– 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– Definition– 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– Definition– 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– Definition– 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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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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x

Y

3i

2j

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

A.j=2

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A.B=(x1.y1+x2.y2+……xn.yn)

f.g=?

Inner Product

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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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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

– 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)– Definition– Inverse DFS– Relation between DFS and FS– Discrete Convolution

• Discrete-Time Fourier Transform (DTFT)– Definition– 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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