# Analog and Digital Signals, Time and Frequency Representation of Signals

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1 1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic

2 2 Data vs. Signal Analog vs. Digital Analog Signals Simple Analog Signals Composite Analog Signals Digital Signals

3 Data vs. Signal 3 Data information formatted in human/machine readable form examples: voice, music, image, file Signal electric or electromagnetic representation of data transmission media work by conducting energy along a physical path; thus, to be transmitted, data must be turned into energy in the form of electro-magnetic signals Transmission communication of data through propagation and processing of signals Idea, Feeling, Knowledge Data Coding Signal Signal Decoding Data Sender Channel or Communication Medium Receiver

4 Signal Representation 4 Signal Representation typically in 2D space, as a function of time, space or frequency when horizontal axis is time, graph displays the value of a signal at one particular point in space as a function of time when horizontal axis is space, graph displays the value of a signal at one particular point in time as a function of space time space The time- and space- representation of a signal often resemble each other, though the signal envelope in the space-representation is different (signal attenuates over distance).

5 Signal Representation (cont.) 5 Example [ signal in time and space ]

6 6 Data vs. Signal Analog vs. Digital Analog Signals Simple Analog Signals Composite Analog Signals Digital Signals

7 Analog vs. Digital 7 Analog vs. Digital Data analog data representation variable takes on continuous values in some interval, e.g. voice, temperature, etc. digital data representation variable takes on discrete (a finite & countable number of) values in a given interval, e.g. text, digitized images, etc. Analog vs. Digital Signal analog signal analog signal signal that is continuous in time and can assume an infinite number of values in a given range (continuous in time and value) discrete (digital) signal signal that is continuous in time and assumes only a limited number of values (maintains a constant level and then changes to another constant level)

8 Analog vs. Digital (cont.) 8 Both analog and digital data can be transmitted using either analog or digital signals. example: analog signaling of analog and digital data will talk more about this later

9 Analog vs. Digital (cont.) 9 Periodic vs. Aperiodic Signals periodic signal completes a pattern within some measurable time frame, called a period (T), and then repeats that pattern over subsequent identical periods T R s. t. s(t + T) = s(t), t -, + T - smallest value that satisfies the equation T is (typically) expressed in seconds aperiodic signal changes without exhibiting a pattern that repeats over time t t+t t+2t periodic analog signal periodic digital signal

10 10 Data vs. Signal Analog vs. Digital Analog Signals Simple Analog Signals Composite Analog Signals Digital Signals

11 11

12 Analog Signals 12 Classification of Analog Signals (1) Simple Analog Signal cannot be decomposed into simpler signals sinewave most fundamental form of periodic analog signal mathematically described with 3 parameters s(t) = A sin(2πft + ϕ) (1.1) peak amplitude (A) absolute value of signal s highest intensity unit: volts [V] (1.2) frequency (f) number of periods in one second unit: hertz [Hz] = [1/s] inverse of period (T)! The origin is usually taken as the last previous passage through zero from the negative to the positive direction. (1.3) phase (φ) absolute position of the waveform relative to an arbitrary origin unit: degrees [º] or radians [rad] A T = 1/f 0[s] 1[s] (2) Composite Analog Signal composed of multiple sinewaves

13 Simple Analog Signals 13 Phase in Simple Analog Signals measured in degrees or radians 360º = 2π rad 1º = 2π/360 rad 1 rad = (360/2π)º = phase shift of 360º = shift of 1 complete period phase shift of 180º = shift of 1/2 period phase shift of 90º = shift of 1/4 period 5V 1s φ = 0º or 360º φ = 90º φ = 180º

14 Analog Signals 14 Example [ period and frequency ] Unit Equivalent Unit Equivalent seconds (s) 1 s hertz (Hz) 1 Hz milliseconds (ms) 10 3 s kilohertz (KHz) 10 3 Hz microseconds (μs) 10 6 s megahertz (MHz) 10 6 Hz nanoseconds (ns) 10 9 s gigahertz (GHz) 10 9 Hz picoseconds (ps) s terahertz (THz) Hz units of period and respective frequency (a) Express a period of 100 ms in microseconds. 100 ms = s = μs = 10 5 μs (b) Express the corresponding frequency in kilohertz. 100 ms = s = 10-1 s f = 1/10-1 Hz = KHz = 10-2 KHz

15 Simple Analog Signals 15 Frequency in Simple Analog Signals rate of signal change with respect to time change in a short span of time high freq. change over a long span of time low freq. signal does not change at all zero freq. signal never completes?? a cycle T= f=0, DC sig. signal changes instantaneously freq. signal completes a cycle?? in T=0 f= Time Domain Plot specifies signal amplitude at each instant of time does NOT express explicitly signal s phase and frequency Frequency Domain Plot specifies peak amplitude with respect to freq. phase CANNOT be shown in the frequency domain

16 Simple Analog Signals 16 One spike in frequency domain shows two characteristics of the signal: spike position = signal frequency, spike height = peak amplitude. Analog signals are best represented in in the frequency domain.

17 Simple Analog Signals 17 Example [ time vs. frequency domain ]

18 Composite Analog Signals 18 Fourier Analysis any composite signal can be represented as a combination of simple sine waves with different frequencies, phases and amplitudes s(t) = A sin(2πf1t + ϕ1) + A 2sin(2πf2t + ϕ2 ) 1 + periodic composite signal (period=t, freq. = f 0 =1/T) can be represented as a sum of simple sines and/or cosines known as Fourier series:... s(t) = A 0 + t) n 0 n 0 2 n= 1 [ A cos(2πnf t) + B sin(2πnf ] With the aid of good table of integrals, it is easy to determine the frequency-domain nature of many signals. A B n n T 2 = s(t)cos(2 nf T π 0 T 2 = s(t)sin(2 nf T π t)dt, n = 0,1,2, t)dt, n = 1,2,3, f 0 is referred to as fundamental frequency integer multiples of f 0 are referred to as harmonics

19 Composite Analog Signals (cont.) 19 Angular Frequency aka radian frequency number of 2π revolutions during a single period of a given signal ω = 2π T = 2π f simple multiple of ordinary frequency s(t) = A 0 + t) n 0 n 0 2 n= 1 [ A cos(nω t) + B sin(nω ] T 2 An = s(t)cos n 0t)dt n = 0,1,2,... T ( ω, 0 T 2 Bn = s(t)sin n 0t)dt n = 1,2,... T ( ω, 0

20 Composite Analog Signals (cont.) 20 Example [ periodic square wave ] No DC component!!! 4A 4A 4A s(t) = sin(2πft) + sin(2π(3f)t) + sin(2π(5f)t) +... π 3π 5π three harmonics adding three harmonics With three harmonics we get an approximation of of a square wave. To get the actual square, all harmonics up to to should be added.

21 Composite Analog Signals (cont.) 21 Example [ composite analog signal ]

22 Composite Analog Signals (cont.) 22 Frequency Spectrum of Analog Signal Absolute Bandwidth of Analog Signal Effective Bandwidth of Analog Signal range (set) of frequencies that signal contains width of signal spectrum: B = f highest -f lowest range of frequencies where signal contains most of its power/energy square wave 3-harmonic representation

23 Composite Analog Signals (cont.) 23 Example [ frequency spectrum and bandwidth of analog signal ] A periodic signal is composed of five sinewaves with frequencies of 100, 300, 500, 700 and 900 Hz. What is the bandwidth of this signal? Draw the frequency spectrum, assuming all components have a max amplitude of 10V. Solution: B = f highest -f lowest = = 800 Hz The spectrum has only five spikes, at 100, 300, 500, 700, and 900.

24 Composite Analog Signals (cont.) 24 Example [ frequency spectrum of a data pulse ] 2Aτ sin πn T T πnτ T ( τ ) sin( π f τ ) π f τ or sin(x) x envelope of frequency spectrum periodic signal discrete freq. spectrum aperiodic signal continuous freq. spectrum What happens if τ 0???

25 Composite Analog Signals (cont.) 25 Composite Signals and Transmission Medium no transmission medium is perfect each medium passes some frequencies and blocks or weakens others composite signal sent at one end of transmiss. medium (comm. channel), may not be received in the same form at the other end passing a square wave through any medium will always deform the signal!!! communication channel Channel Bandwidth range of frequencies passed by the channel difference between highest and lowest frequency that channel can satisfactorily pass

26 26 Data vs. Signal Analog vs. Digital Analog Signals Simple Analog Signals Composite Analog Signals Digital Signals

27 Digital Signals 27 Digital Signals sequence of voltage pulses (DC levels) each pulse represents a signal element binary data are transmitted using only 2 types of signal elements ( 1 = positive voltage, 0 = negative voltage ) key digital-signals terms: bit interval time required to send a single bit, unit: [sec] bit rate number of bit intervals per second unit: [bps] Most digital signals are aperiodic, so it it is is not appropriate // correct to to talk about their period.

28 Digital Signals (cont.) 28 Digital Signal as a Composite Analog Signal digital signal, with all its sudden changes, is actually a composite signal having an infinite number of frequencies a digital signal is a composite signal with an infinite bandwidth if a medium has a wide bandwidth, a digital signal can be sent through it some frequencies will be weakened or blocked; still, enough frequencies will be passed to preserve a decent signal shape what is the minimum required bandwidth B [Hz] of a band-limited medium if we want to send n [bps]?

29 Digital Signals (cont.) 29 Example [ approximation of digital signal s spectrum using 1 st harmonic ] Assume our computer generates 6 bps. Possibilities (periodic combinations) : , , , etc. 1. Best case: min # of changes min freq. of substitute analog signal 2. Worst case max # of changes max freq. of substitute analog signal bit rate: n = 6 [bps] frequency: B = 3 [Hz] n B 2

30 Exercise Before data can be transmitted, they must be transformed to. (a) periodic signals (b) electromagnetic signals (c) aperiodic signals (d) low-frequency sinewaves 2. In a frequency-domain plot, the vertical axis measures the. (a) peak amplitude (b) frequency (c) phase (d) slope 3. In a time-domain plot, the vertical axis measures the. (a) peak amplitude (b) amplitude (c) frequency (d) time 4. If the bandwidth of a signal is 5 KHz and the lowest frequency is 52 KHz, what is the highest frequency. (a) 5 KHz (b) 10 KHz (c) 47 KHz (d) 57 KHz

31 Exercise If one of the components of a signal has a frequency of zero, the average amplitude of the signal. (a) is greater than zero (b) is less than zero (c) is zero (d) (a) or (b) 6. Give two sinewaves A and B, if the frequency of A is twice that of B, then the period of B is that of A. (a) one-half (b) twice (c) the same as (d) indeterminate from 7. A device is sending out data at the rate of 1000 bps. (a) How long does it take to send out 10 bits? (b) How long does it take to send out a single character (8 bits)? (c) How long does it take to send a file of 100,000 characters?

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