DIGITAL SIGNAL PROCESSING - Assoc. Prof. Dr. Nguyen Huu Phuong

UNIVERSITY OF SCIENCE FACULTY OF ELECTRONICS & TELECOMMUNICATION Assoc. Prof. Dr. Nguyen Huu Phuong EDITION SEP/2012 1 CHAPTER 1 DISCRETE – TIME SIGNALSAND SYSTEMS Signals represent information about data, voice, audio, image, video… There are many ways to classify signals but here we categorize signals as either analog (continuous-time) or digital (discrete-time). Signal processing is to use circuits and systems (hardware and software) to act on input signal to give output signal which differs from the input, the way we would like to. Digital systems have many advantages over analog such as noise immunity, easiness of storage and transmission… To convert an analog signal to a digital equivalence we first sample it at regular intervals, quantize the samples, then code the quantized values into binary numbers. If only the sampling is used we obtain discrete – time signals. But to process signals in digital systems (such as computers) we ought to go through all three steps. Usually the last two steps, quantization and binary encoding, are understood, then the terms discrete-time and digital are equivalent and interchangeable. Thus Digital signal processing (DSP) and discrete – time signal processing (DTSP) usually mean the same thing. Besides signals that we honor, there are many unwelcome elements such as noise, interference, jitter … that should be eliminated or minimized. Systems can be just simple logic circuits, simple programmes, up to complex structures including both hardware and software, such as computers. We will discuss various types of digital systems, of which the linear and time invariant (LTI) ones are usually assumed. Typical systems are filters. 1.1 CONTINUOUS – TIME SIGNALS A signal is the variation of an amplitude with time. The amplitude can be a voltage, current, power,... but in circuits and systems the most often used representative is voltage. Continuous – time (also taken to mean analog) signals have their amplitudes varying continuously with time. They are generated by electronic circuits, or by natural sources, such as temperature, voice, video..., and converted to electric signals by sensors or transducers. Signals are often depicted by their waveforms which are the graphical illustrations for easy visualization. 1.1.1 Mathematical representation of signals Instead of describing signals by words or by plotting their waveforms, the more objective and concise way is to express signals mathematically, whenever possible. Mathematical representation of signals in time domain and transform domain is needed for analysis and design of circuits and systems. For example a simple problem in Fig.1.1 cannot be solved just by language description of the signal and the circuit. Input Circuit Output R 560Ω C ? 1Vp –- 1kHz 0,1mF Fig.1.1: What is the output signal ?h Sinusoidal signal Sinusoidal signals (sinusoids or sine waves) are the most popular analog signal (Fig.1.2). They are smooth, easy to generate, and has many properties and applications. The mathematical expression is x(t) = Acos(Ωt + Φo) (1.1) where A is the peak value, Ω angular frequency (radians/s), t time (sec), Φo initial phase (radians), that is, phase at t = 0, Ω = 2πF with F is frequency (Hz), T = 1 / F = 2π/Ω the period (sec). x(t) 2 A T AcosΦ0 0 t –A Fig.1.2: Sinusoidal signal Above expression contains all parameters we need: Amplitude (peak, rms, average), and periodicity (period, frequency). Other waveforms, except the constant value, do not have this compactness. For example, for the symmetric square wave (Fig.1.3) the mathematical expression consists of one part for amplitude, and the other for periodicity: x(t) = –A , T £ t £ 0 (1.2) +A , 0 £ t £ T x(t) = x(t ± nT) , n = 1, 2, 3 … The sinusoid and the square wave are deterministic. For random signals, we cannot, in general, represent them mathematically. Electric noises and interferences are examples of random signals. x(t) A –T/2 0 T/2 T 2T t –A Fig.1.3: Symmetric square wave 1.1.2 Some special signals There are two singular signals often used in circuit analysis and signal processing. (a) Unit impulse: The unit impulse (delta Dirac function) is evolved from a symmetric rectangular pulse of width t and amplitude 1/ t when t → 0 (Fig.1.4). Its mathematical expression is d(t) = ¥, t = 0 0, t ¹ 0 ¥ d(t)dt = 1 (1.3) The latter equation means that the area (or intensity) of the unit impulse is 1(unit). According to this definition, d(–t) = d(t) (1.4) 3 1 t t d(t) Ad(t) d(t-t0) t 0 t t 0 t 0 t 0 t0 t 2 2 (a) (b) (c) (d) Fig.1.4: Unit impulse d(t) When the impulse has intensity of A instead of 1 we write Ad(t) (Fig.1.4c). When the unit impulse is delayed by to we write d(t – to) (Fig.1.4d), then d(t – tO) = ¥ , t = tO (1.5) 0 , t ¹ tO ¥ d(t tO )dt =t2 d(t tO )dt = 1, t1 < tO < t2 A signal x(t) when multiplied with delayed impulse d(t t0 ) is the value x(to) at to: x(t)d(t – to) = x(to) (1.6) (b) Unit step: Fig.1.5 is the unit step. The signal rises suddenly from 0 to 1 at time t = 0 then remains unchanged, similarly to the closure of an electric switch. Its mathematical definition is u(t) = 0, t < 0 1, t ³ 0 (1.7) Au(t) A 1 u(t) A Au(t0 – t) [ 0 t 0 t 0 (a) (b) t0 (c) t Fig.1.5: Unit step The unit impulse d(t) and the unit step u(t) are related as follows: u(t) = t¥ d(t`)dt` = 0, t < 0 1, t ³ 0 (1.8a) d(t) = du(t) (1.8b) 1.1.3 Complex signals Natural physical quantities, signals included, are real-valued. However sometimes the imaginary operator j = 1 is appended to them by reason of mathematical convenience, such as to take into account the phase difference between voltages and currents in AC circuits. Following is an example of a complex signal : x(t) = 5cosWt – j5sinWt (1.9) A complex signal comprises a real and an imaginary part: 4 x(t) = xR (t) + jxI (t) (1.10a) A complex signal can be expressed in terms of its magnitude and phase in the polar coordinate (Fig.1.6) x(t) = xR (t) + jxI (t) = x(t)e jΦ (t) (1.10b) Imaginary x(t) xI (t) x(t) Φ(t) 0 x R (t) Real Fig.1.6: Complex signal and polar coordinate x(t) is the magnitude or modulus, and the phase or phase angle is denoted by Φ(t) or argx(t) or Ð x(t). They are x(t) = x2 (t) + x2 (t) (1.11a) Φ(t) = tan 1 xI (t) (1.11b) R A point to note is that magnitude is an absolute value while amplitude is a signed value, but we do not always need to differentiate the two terms. Example 1.1.1 A complex signal is = 5 Ω − 5 Ω (1.9). Find its real part, imaginary part, magnitude and phase. Solution – Real part: xR (t) = 5cosWt – Imaginary part: xI (t) = –5sinWt 1/2 – Magnitude: x(t) (5 cosΩ t) + ( 5sinΩ t) = 5 – Phase: Φ(t) = tan 1 5sinΩ t = tan ( 1) = –45 5cosΩ t According to this representation we can consider a complex signal as a vector and write as x(t). Imaginary x(t) xI(t) 0 Φ(t) xR(t) = x*R(t) -Φ(t) Real x*I(t) x*(t) Fig.1.7: A complex signal x(t) and its complex conjugate x*(t) ... - tailieumienphi.vn