Ch2.1: Discrete-Time Signals Information source and input transducer Source Coding • Ch 2: DT Signals and Systems Information sink and output transducer Elec3100 Chapter 2 Channel Coding Modulator Questions to be answered: – The Sampling Process: How to represent a continuous-time signal by a discrete-time signal without losing any information. (Why do we want to do this?) – Discrete-Time Signals: Representation (in timedomain), operations, and classifications of discretetime signals. – Typical Sequences: Widely utilized sequences that can be utilized to represent other sequences. – Correlation of Signals: How similar two signals are? How can we utilize this? Source Decoding Channel Decoding Channel Demodulator (Matched Filter) 1 Ch2.1: Discrete-Time Signals • The Sampling Process • Discrete-Time Signals – Time-Domain Representation – Operations on Sequences – Classification of Sequences • Typical Sequences • Correlation of Signals Elec3100 Chapter 2 2 3 4 Sampling: From Continuous to Discrete • Often, a discrete-time sequence š„[š] is developed by uniformly sampling a continuous-time signal š„% š” . • The relation: š„ š = š„% š” |*+,- = š„% šš , š = āÆ , −1,0,1, … • Time variable t is related to time variable n as š”, = šš = , 56 = 78, 96 where š¹- and Ω - denoting the sampling frequency and sampling angular frequency. Elec3100 Chapter 2 http://en.wikipedia.org/wiki/Angular_frequency 5 Sampling Process • Consider the continuous-time signal š„ š” = š“ššš 2ššC š” + š = š“ššš (ΩC š” + š). • The corresponding discrete-time signal is 2šΩC š„ š = š“ššš ΩC šš + š = š“ššš š + š = š“ššš (šC š + š) Ωwhere šC = frequency. 789I 96 =ΩC š is the normalized digital angular • If the unit of T is second, then – Unit of šC is radians/sample – Unit of ΩC is radians/second – Unit of šC is hertz (Hz). Elec3100 Chapter 2 6 Sampling Process: Aliasing • Consider three continuous signals šK š” = ššš 6šš” , š7 š” = ššš 14šš” , šN š” = ššš 26šš” . • Sampling them at a rate of 10Hz generates šK š = ššš 0.6šš , š7 š = ššš 1.4šš , šN š = ššš 2.6šš • Each sequence has exactly the same value for any given n. Elec3100 Chapter 2 7 Sampling Process: Aliasing • This can be verified by observing that š7 š = ššš 1.4šš = cos( 2š − 0.6š š) šN š = ššš 2.6šš = cos( 2š + 0.6š š) • As a result, the three sequences are identical and it is difficult to associate a unique continuous-time function with each of them. • This phenomenon of a continuous-time signal of a higher frequency acquiring the identity of a sinusoidal sequence of a lower frequency after sampling is called aliasing. • How can we solve this problem? Elec3100 Chapter 2 8 Sampling Theorem • Recall šC = 789I . 96 • Thus, if |Ω - | > 2|ΩC |, then the normalized digital angular frequency šC obtained by sampling the parent continuoustime signal will be in the range −š < šC < š. No aliasing • Otherwise, if |Ω - < 2 ΩC |, šC will foldover into a lower digital frequency šC =< 2šΩC /Ω - >78 in the range −š < šC < š. • Thus, to prevent aliasing the sampling frequency š“š» should be greater than 2 times of the frequency š“š . Elec3100 Chapter 2 9 Why sampling is reversible ? 10 11 12 13 14 15 16 17 Ch2.1: Discrete-Time Signals • The Sampling Process • Discrete-Time Signals – Time-Domain Representation – Operations on Sequences – Classification of Sequences • Typical Sequences • Correlation of signals Elec3100 Chapter 2 18 Time-Domain Representation • Discrete-time signals are represented as sequences of numbers, called samples. • Sample value of a typical sequence is denoted as š„[š] with š being an integer. • Discrete-time signal is represented by {š„[š]}. Elec3100 Chapter 2 19 Generating Discrete-Time Signals • In some applications, a discrete-time signal {š„[š]} may be generated by periodically sampling a continuous-time signal š„% (š”) at uniform intervals of time. • The š-th sample is given by š„ š = š„% š” |*+,- = š„% šš • The spacing T is called the sampling interval/period. • Reciprocal of T is called sampling frequency: š¹- = Elec3100 Chapter 2 K . - 20 Real and Complex Sequence • š„ š is called the š-th sample of the sequence, no matter whether {š„[š]} has been obtained by sampling. • {š„[š]} is a real sequence if š„ š is real for all values of n. Otherwise, {š„[š]} is complex. • A complex sequence {š„[š]} can be written as {š„[š]} ={š„Z[ [š]} + j{š„Z[ [š]} where {š„Z[ [š]} and {š„\] [š]} denote the real and imaginary parts. • Normally, braces are ignored to denote a sequence if there is no ambiguity. Elec3100 Chapter 2 21 Example • Let š„ š = cos(0.25š) and y š = šš„š(š0.3š) denote two sequences. – Are they real or complex sequences? – Find the real and imagine parts of both signals. – Find the complex conjugate sequence. Elec3100 Chapter 2 22 Why complex signals? Complex envelope (complex amplitude) u(t) is the low-pass equivalent representation of a bandpass real signal x(t) 23 Spectrum of real BP signal x(t) X(f) U(f) Spectrum of complex envelop u(t) 24 Types of Discrete-Time signals • Sampled-data signals: Samples are continuous-valued. • Digital signals: Samples are discrete-valued. • Digital signals are normally obtained by quantizing the sample values by rounding or truncation. Elec3100 Chapter 2 25 Length of Discrete-Time signals • A discrete-time signal may be a finite-length or infinitelength sequence. • Finite-length sequence is only defined for finite time interval: −∞ < šK≤ š ≤ š7 < ∞ where the length is š = š7 − šK + 1. A length-N sequence is referred to as a Npoint sequence. • Example: š„ š = š7, −3 ≤ š ≤ 4, what is the length? • The length of a finite-length sequence can be increased by zero-padding, i.e., by appending it with zeros. š7, −3 ≤ š ≤ 4 • Example: š„ š = g , what is the length? 0, 5 ≤ š ≤ 8 Elec3100 Chapter 2 26 Ch2.1: Discrete-Time Signals • The Sampling Process • Discrete-Time Signals – Time-Domain Representation – Operations on Sequences – Classification of Sequences • Typical Sequences • Correlation of signals Elec3100 Chapter 2 27 Operations on Sequences • Purpose: Develop another sequence with more desirable properties. • Example: – Input signal: Signal corrupted by additive noise – Design a discrete-time system to generate an output by removing the noise component. Elec3100 Chapter 2 28 Basic Operations • Product (Modulation) operation: • Application: Develop a finite-length sequence from an infinite-length sequence by multiplying the latter with a finite-length called the Window Sequence. The process is called Windowing. • Addition operation Elec3100 Chapter 2 29 Basic Operations • Multiplication operation: • Time-shifting operation: š¦ š = š„[š − š], where N is an integer. – N>0, delaying operation – N<0, advance operation. • Time reversal (folding) operation: š¦ š = š„[−š]. • Branching operation: Provide multiple copies of a sequence. Elec3100 Chapter 2 30 Basic Operations: Examples • Example 1: Consider two sequences š š = 3, 4, 6, −9, 0 , 0 ≤ š ≤ 4 š š = 2, −1, 4, 5, −3 , 0 ≤ š ≤ 4 Determine the new sequence generated from Product, Addition, and Multiplication (A=2). • Example 2: Consider two sequences š š = 3, 4, 6, −9, 0 , 0 ≤ š ≤ 4 š š = −2, 1, −3 , 0 ≤ š ≤ 2 Determine the new sequence generated from Product, Addition, and Multiplication (A=0.5). Elec3100 Chapter 2 31 Basic Operations: Ensemble Averaging • Signal model: Let š \ denote the noise vector corrupting the i-th measurement š± š = š¬ + šš • Operation: Averaging over K measurements š± %o[ r r \+K \+K 1 1 = q š± \ = š¬ + q š\ š¾ š¾ • Purpose: Improving the quality of measured data corrupted by additive random noise. • Assumption: The measured data remains essentially the same from one measurement to next, while additive noise is random. Elec3100 Chapter 2 32 Basic Operations: Ensemble Averaging Elec3100 Chapter 2 33 Combinations of Basic Operations What are the basic operations utilized? Elec3100 Chapter 2 34 Sampling Rate Alteration • Definition: Generating a new sequence š¦[š] with a sampling rate šwš» higher or lower than that of the sampling rate šš» of a given sequence š„[š]. • Sampling rate alteration ratio š = šyš» šš» • If š > 1, the process is called interpolation. – Interpolator = up-sampler + DT system • If š < 1, the process is called decimation. – Decimator = DT system + down-sampler Elec3100 Chapter 2 35 Operations: Up-Sampling • In up-sampling by an integer factor of L>1, L-1 equidistant zero-valued samples are inserted by the upsampler between each two consecutive samples. Elec3100 Chapter 2 36 Operations: Down-Sampling • In down-sampling by an integer factor of M>1, every M-th samples of the input sequence are kept and the other samples are removed What kinds of DT systems do we need? Why? Elec3100 Chapter 2 37 Ch2.1: Discrete-Time Signals • The Sampling Process • Discrete-Time Signals – Time-Domain Representation – Operations on Sequences – Classification of Sequences • Typical Sequences • Correlation of signals Elec3100 Chapter 2 38 Sequence Classification: Symmetry • Conjugate-symmetric sequence: š„ š = š„ ∗[−š]. What if it is real? • Conjugate-antisymmetric sequence: š„ š = −š„ ∗[−š]. If real? • What is š„ 0 for different cases? Elec3100 Chapter 2 39 Representing a Complex Sequence • Any complex sequence can be expressed as a sum of its conjugate-symmetric part and its conjugate-antisymmetric part: š„ š = š„{| š +š„{% š where š„{| š„{% 1 š = (š„ š + š„ ∗ [−š]) 2 1 š = (š„ š − š„ ∗ [−š]) 2 • What about a real sequence? • Example: Consider a length-7 sequence defined for −3 ≤ š ≤ 3, š š = 0,1 + š4, −2 + š3,4 − š2, −5 − š6, −š2,3 . Determine its conjugate sequence, and its conjugatesymmetric and conjugate-antisymmetric parts. Elec3100 Chapter 2 40 Finite-Length Sequence • A length-N sequence š„ š , 0 ≤ š ≤ š − 1, can be expressed as š„ š = š„}{| š +š„}{% š where 1 š„}{| š = š„ š + š„ ∗ < −š >~ , 0 ≤ š ≤ š − 1, 2 1 š„}{% š = š„ š − š„ ∗ < −š >~ , 0 ≤ š ≤ š − 1, 2 are the periodic conjugate-symmetric and periodic conjugate-antisymmetric parts, respectively. • For a real sequence, they are called the periodic even š„}[ š and periodic odd parts š„}C [š], respectively. Elec3100 Chapter 2 41 Example • Consider a length-4 sequence defined for 0 ≤ š ≤ 3 š¢ š = {1 + š4, −2 + š3,4 − š2, −5 − š6}. Determine its conjugate-symmetric part. • Solution: Its conjugate sequence is given by š¢∗ š = {1 − š4, −2 − š3,4 + š2, −5 + š6}. Next, determine the modulo-4 time-reversed version š¢∗ < −š >• = š¢∗ −š + 4 For example, š¢∗ < −0 >• = š¢∗ 0 = 1 − š4 Thus, š¢∗ < −š >• = {1 − š4, −5 + š6,4 + š2, −2 − š3} 1 Finally, š„}{| š = š„ š + š„ ∗ < −š >~ 2 Elec3100 Chapter 2 42 Periodic Conjugate-Symmetric • A length-N sequence š„ š , 0 ≤ š ≤ š − 1, is called a periodic conjugate-symmetric sequence if š„ š = š„ ∗ < −š >~ = š„ ∗ < š − š >~ and is called a periodic conjugate-antisymmetric sequence if š„ š = −š„ ∗ < −š >~ = −š„ ∗ < š − š >~ Elec3100 Chapter 2 43 Periodic Sequence • A sequence š„‹ š satisfying š„‹ š = š„‹ š + šš is called a periodic with period N where N is a positive integer and k is any integer. • The smallest value of N is called the fundamental period. • A sequence not satisfying the periodicity condition is called an aperiodic sequence. Elec3100 Chapter 2 44 Energy and Power Signals • Total energy of a sequence š„ š is defined by • ā°Å½ š = q |š„[š]|7 ,+•• • An infinite-length sequence with finite sample values may or may not have finite energy. A finite-length sequence with finite sample values has finite energy. • The average power of an aperiodic sequence is K 7. defined as šÅ½ = lim 7r”K ∑r |š„[š]| ,+•r r→• • The average power of a periodic sequence š„‹ š with a K ~•K 7. period N is given by šÅ½ = ~ ∑,+– |š„[š]| ‹ Elec3100 Chapter 2 45 Example • Example: Consider the causal sequence š„ š defined by 3(−1), , š ≥ 0 š„ š =g 0, š < 0 Determine the energy and power. • Solution: Energy = ? Its average power is given by r 1 9(š¾ + 1) šÅ½ = lim (9 q 1) = lim = 4.5 r→• 2š¾ + 1 r→• 2š¾ + 1 ,+– Elec3100 Chapter 2 46 Energy and Power Signals • An infinite energy signal with finite average power is called a power signal. • Example- A periodic sequence which has a finite average power but infinite energy. • A finite energy signal with zero average power is called an energy signal. • Example- A finite-length sequence which has finite energy but zero average power. Elec3100 Chapter 2 47 Ch2.1: Discrete-Time Signals • The Sampling Process • Discrete-Time Signals – Time-Domain Representation – Operations on Sequences – Classification of Sequences • Typical Sequences • Correlation of signals Elec3100 Chapter 2 48 Basic Sequences • Unit sample sequence š„ š = g • Unit step sequence: š¢ š = g Elec3100 Chapter 2 1, š = 0 0, š ≠ 0 1, š ≥ 0 0, š < 0 49 Basic Sequences Elec3100 Chapter 2 50 Basic Sequences Elec3100 Chapter 2 51 Basic Sequences Elec3100 Chapter 2 52 Basic Sequences Elec3100 Chapter 2 53 Basic Sequences Elec3100 Chapter 2 54 Basic Sequences Elec3100 Chapter 2 55 Basic Sequences Elec3100 Chapter 2 56 Basic Sequences Elec3100 Chapter 2 57 Basic Sequences Elec3100 Chapter 2 58 Basic Sequences • Property 1 – Consider š„ š = exp(ššKš) and š¦ š = exp(šš7š) with 0 < šK < š and 2š < š7 < 2š(š + 1) where k is any positive integer. If š7 = šK + 2šš, then š„ š =š¦ š . • Because of Property 1, a frequency šC in the neighborhood of ω = 2šš is indistinguishable from a frequency š– − 2šš in the neighborhood of š = š and a frequency in the neighborhood of ω = š(2š + 1) is indistinguishable from a frequency š– − 2šš in the neighborhood of š = š . High or Low frequency? Elec3100 Chapter 2 59 Basic Sequences • Property 2- The frequency of oscillation of Acos(šC š) increases as šC increases from 0 to š , and then decreases as šC increases from š to 2š. Thus, frequencies in the neighborhood of š = 0 are called low frequencies, and frequencies in the neighborhood of š = š are called high frequencies. • Frequencies in the neighborhood of š = 2šš are called low frequencies, and frequencies in the neighborhood of š = š(2š + 1) are called high frequencies. • Example: Which is a low-frequency signal? Elec3100 Chapter 2 60 Basic Sequences Elec3100 Chapter 2 61 Ch2.1: Discrete-Time Signals • The Sampling Process • Discrete-Time Signals – Time-Domain Representation – Operations on Sequences – Classification of Sequences • Typical Sequences • Correlation of signals Elec3100 Chapter 2 62 Correlation of Signals • There are applications where it is necessary to compare one reference signal with one or more signals to determine the similarity between the pair and to determine additional information based on the similarity. • For example, in digital communications, a set of data symbols are represented by a set of unique discrete-time sequences. • If one of these sequences has been transmitted, the receiver has to determine which particular sequence has been received by comparing the received signal with every member of possible sequences from the set. Elec3100 Chapter 2 63 Correlation of Signals • Similarly, in radar and sonar applications, the received signal reflected from the target is a delayed version of the transmitted signal and by measuring the delay, one can determine the location of the target. • The detection problem gets more complicated in practice, as often the received signal is corrupted by additive random noise. Elec3100 Chapter 2 64 Correlation of Signals • A measure of similarity between a pair of energy signals, š„[š] and š¦[š], is given by the cross-correlation sequence defined by • šÅ½Ÿ š = q š„ š š¦ š − š , š = 0, ±1, ±2,... ,+•• • The parameter š is called lag, indicating the time-shift between the pair of signals. The ordering in the subscripts š„š¦ specifies that š„ š is the reference while š¦ š being shifted with respect to š„ š . • We can obtain • • šŸÅ½ š = q š¦ š š„ š − š = q š¦ š + š š„ š = šÅ½Ÿ −š ,+•• Elec3100 Chapter 2 ]+•• 65 Correlation of Signals • The autocorrelation sequence of š„[š] is given by • šÅ½Å½ š = q š„ š š„ š − š , š = 0, ±1, ±2,... ,+•• 7 • Note: šÅ½Å½ 0 = ∑• ,+•• š„ š denotes the energy of š„[š]. • From šŸÅ½ š = šÅ½Ÿ −š , we know šÅ½Å½ š = šÅ½Å½ −š , implying that šŸÅ½ š is an even function for real š„[š]. • Question: Can you find the similarity between correlation and convolution? Elec3100 Chapter 2 66 Normalized Forms of Correlation Elec3100 Chapter 2 67 Correlation for Power Signals Elec3100 Chapter 2 68 Correlation for Periodic Signals Elec3100 Chapter 2 69