# PYKC Jan Lecture 1 Slide 1

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1 Aims and Objectives E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London! By the end of the course, you would have understood: Basic signal analysis (mostly continuous-time) Basic system analysis (also mostly continuous systems) Time-domain analysis (including convolution) Laplace Transform and transfer functions Fourier Series (revision) and Fourier Transform Sampling Theorem and signal reconstructions Basic z-transform URL: Lecture 1 Slide 1 Lecture 1 Slide 2 About the course A demonstration..! Lectures - around 9 weeks (15-17 hours)! Problem Classes 1 hr per week! Official Hours 2 hrs per week (taken by Dr Naylor)! Assessment 100% examination in June! Handouts in the form of PowerPoint slides! Text Book B.P. Lathi, Linear Systems and Signals, 2 nd Ed., Oxford University Press (~ 36)! This is what you will be able to do in your 3 rd year (helped by this course)! You will be able to design and implement a NOISE CANCELLING system Lecture 1 Slide 3 Lecture 1 Slide 4

2 Examples of signals (1) Lecture 1! Electroencephalogram (EEG) signal (or brainwave) Basics about Signals (Lathi ) Peter Cheung Department of Electrical & Electronic Engineering Imperial College London URL: Lecture 1 Slide 5 Lecture 1 Slide 6 Examples of signals (2) Examples of signals (3)! Stock Market data as signal (time series)! Magnetic Resonance Image (MRI) data as 2-dimensional signal Lecture 1 Slide 7 Lecture 1 Slide 8

3 Size of a Signal x(t) (1) Size of a Signal x(t) (2)! Measured by signal energy E x :! If amplitude of x(t) does not! 0 when t! ", need to measure power Px instead:! Generalize for a complex valued signal to:! Again, generalize for a complex valued signal to:! Energy must be finite, which means Lathi Section 1.1 L1.1 L1.1 Lecture 1 Slide 9 Lecture 1 Slide 10 Size of a Signal x(t) (3) Useful Signal Operations Time Shifting (1)! Signal with finite energy (zero power)! Signal may be delayed by time T:! or advanced by time T:! Signal with finite power (infinite energy)! (t T) = x (t) L1.1 L1.2.1 Lecture 1 Slide 11 Lecture 1 Slide 12

4 Useful Signal Operations Time Scaling (2) Useful Signal Operations Time Reversal (3)! Signal may be compressed in time (by a factor of 2):! Signal may be reflected about the vertical axis (i.e. time reversed):! or expanded in time (by a factor of 2):! Same as recording played back at twice and half the speed respectively L1.2.2! We can combine these three operations.! For example, the signal x(2t - 6) can be obtained in two ways; Delay x(t) by 6 to obtain x(t - 6), and then time-compress this signal by factor 2 (replace t with 2t) to obtain x (2t - 6). Alternately, time-compress x (t) by factor 2 to obtain x (2t), then delay this signal by 3 (replace t with t - 3) to obtain x (2t - 6). L1.2.3 Lecture 1 Slide 13 Lecture 1 Slide 14 Signals Classification (1)! Signals may be classified into: 1. Continuous-time and discrete-time signals 2. Analogue and digital signals 3. Periodic and aperiodic signals 4. Energy and power signals 5. Deterministic and probabilistic signals 6. Causal and non-causal 7. Even and Odd signals Signal Classification (2) Continuous vs Discrete! Continuous-time! Discrete-time Lecture 1 Slide 15 Lecture 1 Slide 16

5 Signal Classification (3) Analogue vs Digital Signal Classification (4) Periodic vs Aperiodic Digital, continuous! A signal x(t) is said to be periodic if for some positive constant T o Analogue, continuous! The smallest value of T o that satisfies the periodicity condition of this equation is the fundamental period of x(t). Analogue, discrete Digital, discrete Lecture 1 Slide 17 Lecture 1 Slide 18 Signal Classification (5) Deterministic vs Random Signal Classification (6) Causal vs Non-causal Deterministic Causal Random Non-causal Lecture 1 Slide 19 Lecture 1 Slide 20

6 Signal Classification (7) Even vs Odd Signal Models (1) Unit Step Function u(t) Even! Step function defined by: Odd! Useful to describe a signal that begins at t = 0 (i.e. causal signal).! For example, the signal represents an everlasting exponential that starts at t = -".! The causal for of this exponential can be described as: L1.4.1 Lecture 1 Slide 21 Lecture 1 Slide 22 Signal Models (2) Pulse signal Signal Models (3) Unit Impulse Function #(t)! A pulse signal can be presented by two step functions:! First defined by Dirac as: Unit Impulse Approximation of an Impulse L1.4.1 L1.4.2 Lecture 1 Slide 23 Lecture 1 Slide 24

7 Signal Models (4) Unit Impulse Function #(t) Multiplying a function \$(t) by an Impulse! May use functions other than a rectangular pulse. Here are three example functions:! Note that the area under the pulse function must be unity! Since impulse is non-zero only at t = 0, and \$(t) at t = 0 is \$(0), we get: Exponential Triangular Gaussian! We can generalise this for t = T: L1.4.2 L1.4.2 Lecture 1 Slide 25 Lecture 1 Slide 26 Sampling Property of Unit Impulse Function The Exponential Function e st (1)! Since we have:! It follows that:! This exponential function is very important in signals & systems, and the parameter s is a complex variable given by:! This is the same as sampling \$(t) at t = 0.! If we want to sample \$(t) at t = T, we just multiple \$(t) with! This is called the sampling or sifting property of the impulse. L1.4.2 L1.4.3 Lecture 1 Slide 27 Lecture 1 Slide 28

8 The Exponential Function e st (2) The Exponential Function e st (2)! If % = 0, then we have the function, which has a real frequency of &! Therefore the complex variable s = " + j# is the complex frequency! The function e st can be used to describe a very large class of signals and functions. Here are a number of example: L1.4.3 L1.4.3 Lecture 1 Slide 29 Lecture 1 Slide 30 The Complex Frequency Plane s = " + j# Even and Odd functions (1) +j# The s-plane! A real function x e (t) is said to be an even function of t if s on y-axis s on right of y-axis -" +"! A real function x o (t) is said to be an odd function of t if s on left of y-axis -j# s on x-axis L1.4.3 L1.5 Lecture 1 Slide 31 Lecture 1 Slide 32

9 Even and Odd functions (2)! Even and odd functions have the following properties: Even x Odd = Odd Odd x Odd = Even Even x Even = Even Even and Odd functions (3)! Consider the causal exponential function! Every signal x(t) can be expressed as a sum of even and odd components because: L1.5 L1.5 Lecture 1 Slide 33 Lecture 1 Slide 34 Relating this lecture to other courses! The first part of this lecture on signals has been covered in this lecture was covered in the 1 st year Communications course (lectures 1-3)! This is mostly an introductory and revision lecture Lecture 1 Slide 35

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