(Part one: Continuous) Signals and Systems (Part one: Continuous)

General Information Lecturer Reference Dr. Yasmine Fahmy Signals and systems, Oppenheim A.V., Wilski A.S., Prentice Hall, 1997 Dr. Yasmine Fahmy

Course Contents Signals (definition, properties , important signals) 2 Systems (definition, properties) Linear Time Invariant (LTI) Systems Fourier Series Fourier Transform LTI Systems in Frequency Domain Applications (Filters, Sampling, Modulation) 2 1 9 Dr. Yasmine Fahmy

variables carrying information Signals variables carrying information Electrical signals Voltages and currents in a circuit Acoustic signals Acoustic pressure (sound) over time Mechanical signals Velocity of a car over time Video signals Intensity level of a pixel (camera, video) over time Dr. Yasmine Fahmy

Continuous / Discrete Velocity Voltage Pixels Daily stock price x(t) t x[n] n -3 -2 -1 0 1 2 3 4 Velocity Voltage Pixels Daily stock price Dr. Yasmine Fahmy

Analog / Digital Continuous Analog Signal t Discrete Analog Signal Continuous Quantized Analog Signal n -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 Dr. Yasmine Fahmy

Analog / Digital Sampling Quantization Coding t n n 01001111010011010 -3 -2 -1 0 1 2 3 4 n -3 -2 -1 0 1 2 3 4 Sampling Quantization Coding -3 -2 -1 0 1 2 3 4 Dr. Yasmine Fahmy

Properties of Signals Signal Energy and Power Transformation in Time (Shift, Reverse, Scaling) Periodic Signals Even and Odd Signals Dr. Yasmine Fahmy

Signal Energy and Power Energy over time interval Average Power over time interval Dr. Yasmine Fahmy

Signal Energy and Power Total Energy Average Power Dr. Yasmine Fahmy

Transformation in Time x(t) Time Shift t x(t+to) x(t-to) t t -to +to +to Advance -to Delay Dr. Yasmine Fahmy

Transformation in Time Time Reverse x(t) x(-t) t t Dr. Yasmine Fahmy

Transformation in Time x(t) Time Scaling t x(׀α׀ t) x(׀α׀ t) t t ׀α ׀ < 1 Compressed ׀α > ׀ 1 Stretched Dr. Yasmine Fahmy

Example 1 Find: The equation of x(t) The values of x(t+1) x(-t+1) , , , Dr. Yasmine Fahmy

NOTE Power signals: Energy signals: Infinite Energy Finite Energy Finite Power Energy signals: Finite Energy Zero Power Dr. Yasmine Fahmy

(is the minimum value of T) Periodic Signals x(t) = x(t+T) Where Period := T Fundamental Period := To (is the minimum value of T) t Dr. Yasmine Fahmy

Example 2 Find the period of the following signals: Dr. Yasmine Fahmy

Even & Odd Signals Even Odd x(t) = x(-t) x(t) = -x(-t) Symmetric around the axis Odd x(t) = -x(-t) Symmetric around the origin t t Dr. Yasmine Fahmy

Even & Odd Signals For any signal x(t) x(t) = xe(t)+ xo(t) Where xe(t)=1/2 [x(t)+x(-t)] xo(t)=1/2 [x(t) -x(-t)] Dr. Yasmine Fahmy

Example 3 Find and Sketch The Even and Odd components of x(t) X(t) 1 , , -1 0 1 Find and Sketch The Even and Odd components of x(t) , Dr. Yasmine Fahmy

Example 3 Dr. Yasmine Fahmy

Lecture Overview Signal (continuous/discrete/analog/digital) Signal Properties Signal Energy and Power Transformation in Time (Shift, Reverse, Scaling) Periodic Signals Even and Odd Signals Dr. Yasmine Fahmy