Lecture 1 - Introduction to Signals.pdf
- 1. Lecture 1 Slide 1 PYKC Jan-7-10 E2.5 Signals & Linear Systems E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London URL: www.ee.ic.ac.uk/pcheung/ E-mail: p.cheung@ic.ac.uk Lecture 1 Slide 2 PYKC Jan-7-10 E2.5 Signals & Linear Systems ! 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 Aims and Objectives Lecture 1 Slide 3 PYKC Jan-7-10 E2.5 Signals & Linear Systems About the course ! 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”, 2nd Ed., Oxford University Press (~£36) Lecture 1 Slide 4 PYKC Jan-7-10 E2.5 Signals & Linear Systems A demonstration ….. ! This is what you will be able to do in your 3rd year (helped by this course) ! You will be able to design and implement a NOISE CANCELLING system
- 2. Lecture 1 Slide 5 PYKC Jan-7-10 E2.5 Signals & Linear Systems Lecture 1 Basics about Signals (Lathi 1.1-1.5) Peter Cheung Department of Electrical & Electronic Engineering Imperial College London URL: www.ee.imperial.ac.uk/pcheung/ E-mail: p.cheung@imperial.ac.uk Lecture 1 Slide 6 PYKC Jan-7-10 E2.5 Signals & Linear Systems Examples of signals (1) ! Electroencephalogram (EEG) signal (or brainwave) Lecture 1 Slide 7 PYKC Jan-7-10 E2.5 Signals & Linear Systems Examples of signals (2) ! Stock Market data as signal (time series) Lecture 1 Slide 8 PYKC Jan-7-10 E2.5 Signals & Linear Systems Examples of signals (3) ! Magnetic Resonance Image (MRI) data as 2-dimensional signal
- 3. Lecture 1 Slide 9 PYKC Jan-7-10 E2.5 Signals & Linear Systems Size of a Signal x(t) (1) ! Measured by signal energy Ex: ! Generalize for a complex valued signal to: ! Energy must be finite, which means L1.1 Lathi Section 1.1 Lecture 1 Slide 10 PYKC Jan-7-10 E2.5 Signals & Linear Systems Size of a Signal x(t) (2) ! If amplitude of x(t) does not ! 0 when t ! ", need to measure power Px instead: ! Again, generalize for a complex valued signal to: L1.1 Lecture 1 Slide 11 PYKC Jan-7-10 E2.5 Signals & Linear Systems Size of a Signal x(t) (3) ! Signal with finite energy (zero power) ! Signal with finite power (infinite energy) L1.1 Lecture 1 Slide 12 PYKC Jan-7-10 E2.5 Signals & Linear Systems Useful Signal Operations –Time Shifting (1) ! Signal may be delayed by time T: ! or advanced by time T: L1.2.1 ! (t – T) = x (t)
- 4. Lecture 1 Slide 13 PYKC Jan-7-10 E2.5 Signals & Linear Systems Useful Signal Operations –Time Scaling (2) ! Signal may be compressed in time (by a factor of 2): ! or expanded in time (by a factor of 2): L1.2.2 ! Same as recording played back at twice and half the speed respectively Lecture 1 Slide 14 PYKC Jan-7-10 E2.5 Signals & Linear Systems Useful Signal Operations –Time Reversal (3) ! Signal may be reflected about the vertical axis (i.e. time reversed): L1.2.3 ! 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). Lecture 1 Slide 15 PYKC Jan-7-10 E2.5 Signals & Linear Systems 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 L1.3 Lecture 1 Slide 16 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Classification (2) – Continuous vs Discrete L1.3 ! Continuous-time ! Discrete-time
- 5. Lecture 1 Slide 17 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Classification (3) – Analogue vs Digital L1.3 Analogue, continuous Digital, continuous Analogue, discrete Digital, discrete Lecture 1 Slide 18 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Classification (4) – Periodic vs Aperiodic L1.3 ! A signal x(t) is said to be periodic if for some positive constant To ! The smallest value of To that satisfies the periodicity condition of this equation is the fundamental period of x(t). Lecture 1 Slide 19 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Classification (5) – Deterministic vs Random Deterministic Random L1.3 Lecture 1 Slide 20 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Classification (6) – Causal vs Non-causal Causal Non-causal
- 6. Lecture 1 Slide 21 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Classification (7) – Even vs Odd Even Odd Lecture 1 Slide 22 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Models (1) – Unit Step Function u(t) L1.4.1 ! Step function defined by: ! 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: Lecture 1 Slide 23 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Models (2) – Pulse signal L1.4.1 ! A pulse signal can be presented by two step functions: Lecture 1 Slide 24 PYKC Jan-7-10 E2.5 Signals & Linear Systems Signal Models (3) – Unit Impulse Function #(t) L1.4.2 ! First defined by Dirac as: Unit Impulse Approximation of an Impulse
- 7. Lecture 1 Slide 25 PYKC Jan-7-10 E2.5 Signals & Linear Systems L1.4.2 ! May use functions other than a rectangular pulse. Here are three example functions: ! Note that the area under the pulse function must be unity Exponential Triangular Gaussian Signal Models (4) – Unit Impulse Function #(t) Lecture 1 Slide 26 PYKC Jan-7-10 E2.5 Signals & Linear Systems Multiplying a function $(t) by an Impulse ! Since impulse is non-zero only at t = 0, and $(t) at t = 0 is $(0), we get: ! We can generalise this for t = T: L1.4.2 Lecture 1 Slide 27 PYKC Jan-7-10 E2.5 Signals & Linear Systems Sampling Property of Unit Impulse Function ! Since we have: ! It follows that: ! 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 Lecture 1 Slide 28 PYKC Jan-7-10 E2.5 Signals & Linear Systems The Exponential Function est (1) ! This exponential function is very important in signals & systems, and the parameter s is a complex variable given by: L1.4.3
- 8. Lecture 1 Slide 29 PYKC Jan-7-10 E2.5 Signals & Linear Systems The Exponential Function est (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 est can be used to describe a very large class of signals and functions. Here are a number of example: L1.4.3 Lecture 1 Slide 30 PYKC Jan-7-10 E2.5 Signals & Linear Systems The Exponential Function est (2) L1.4.3 Lecture 1 Slide 31 PYKC Jan-7-10 E2.5 Signals & Linear Systems The Complex Frequency Plane s = " + j# L1.4.3 The s-plane +j# -j# +" -" s on y-axis s on left of y-axis s on right of y-axis s on x-axis Lecture 1 Slide 32 PYKC Jan-7-10 E2.5 Signals & Linear Systems Even and Odd functions (1) L1.5 ! A real function xe(t) is said to be an even function of t if ! A real function xo(t) is said to be an odd function of t if
- 9. Lecture 1 Slide 33 PYKC Jan-7-10 E2.5 Signals & Linear Systems 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 ! Every signal x(t) can be expressed as a sum of even and odd components because: L1.5 Lecture 1 Slide 34 PYKC Jan-7-10 E2.5 Signals & Linear Systems Even and Odd functions (3) ! Consider the causal exponential function L1.5 Lecture 1 Slide 35 PYKC Jan-7-10 E2.5 Signals & Linear Systems Relating this lecture to other courses ! The first part of this lecture on signals has been covered in this lecture was covered in the 1st year Communications course (lectures 1-3) ! This is mostly an introductory and revision lecture