Course Information Lecture 1vvakilian/CourseECE423/... · Signals and Systems! A signal is a real...

Course InformationLecture 1

ECE 423 Digital Communications

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Exam

Types of Information

! Major classification of data: analog vs. digital

! Analog signals! speech (but words are discrete)

! music (closer to a continuous signal)

! temperature readings, barometric pressure, wind speed

! images stored on film

! Analog signals can be represented (approximately) using bits

! digitized images (can be compressed using JPEG)

! digitized video (can be compressed to MPEG)

! Bits: text, computer data

! Analog signals can be converted into bits by quantizing/digitizing

The word bit (binary digit) was coined in the late 1940s by John Tukey. Today a byte is8 bits. Originally it depended on the computer—6, 9, or 10 bits were also used.

EE 179, April 2, 2014 Lecture 2, Page 2

Analog Messages

! Early analog communication! telephone (1876)

! phonograph (1877)

! film soundtrack (1923, Lee De Forest, Joseph Tykocinski-Tykociner)

! Key to analog communication is the amplifier (1908, Lee De Forest,triode vacuum tube)

! Broadcast radio (AM, FM) is still analog

! Broadcast television was analog until 2009

EE 179, April 2, 2014 Lecture 2, Page 3

Digital Messages

! Early long-distance communication was digital! semaphores, white flag, smoke signals, bugle calls, telegraph

! Teletypewriters (stock quotations)! Baudot (1874) created 5-unit code for alphabet. Today baud is a unit

meaning one symbol per second.

! Working teleprinters were in service by 1924 at 65 words per minute

! Fax machines: Group 3 (voice lines) and Group 4 (ISDN)

! In 1990s the accounted for majority of transPacific telephone use. Sadly,fax machines are still in use.

! First fax machine was Alexander Bains 1843 device required conductive ink

! Pantelegraph (Caselli, 1865) set up telefax between Paris and Lyon

! Ethernet, Internet

There is no name for the unit bit/second. I have proposed claude.

EE 179, April 2, 2014 Lecture 2, Page 4

Analog vs. Digital Systems

! Analog signalsValues varies continously

! Digital signalsValue limited to a finite setDigital systems are more robust

! Binary signalsHave 2 possible valuesUsed to represent bit valuesBit time T needed to send 1 bitData rate R = 1/T bits persecond

EE 179, April 2, 2014 Lecture 2, Page 5

Sampling and Quantization, I

To transmit analog signals over a digital communication link, we mustdiscretize both time and values.

Quantization spacing is2mp

L; sampling interval is T , not shown in figure.

EE 179, April 2, 2014 Lecture 2, Page 6

Sampling and Quantization, II

! Usually sample times are uniformly spaced. Higher frequency contentrequires faster sampling. (Soprano must be sampled twice as fast as atenor.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−0.2

−0.1

0

0.1

0.2

! Values can be uniformly spaced, but nonuniform (logarithmic) spacing isoften used for voice.

EE 179, April 2, 2014 Lecture 2, Page 7

Communication System Block Diagram (Basic)

! Source encoder converts message into message signal (bits)

! Transmitter converts message signal into format appropriate for channeltransmission (analog/digital signal)

! Channel conveys signal but may introduce attenuation, distortion, noise,interference

! Receiver decodes received signal back to message signal

! Source decoder decodes message signal back into original message

EE 179, April 2, 2014 Lecture 2, Page 11

Communication System Block Diagram (Advanced)

EncoderChannel ModulatorEncrypt

DemodulatorDecrypt DecoderChannel

SourceEncoder

Sink

Source

SourceDecoder

NoiseChannel

! Source encoder compresses message to remove redundancy

! Encryption protects against eavesdroppers and false messages

! Channel encoder adds redundancy for error protection

! Modulator converts digital inputs to signals suitable for physical channel

EE 179, April 2, 2014 Lecture 2, Page 12

Signals and Systems

! A signal is a real (or complex) valued function of one or more realvariables.

! voltage across a resistor or current through inductor! pressure at a point in the ocean! amount of rain at 37.4225N, 122.1653W! amount of rain at 16:00 UTC as function of latitude,longitude! price of Google stock at end of each trading day

In this course the independent variable is almost always time.

! Physical signals have units, e.g., volts or psi (Si pascal = N/m2)

! Signals can (usually or in principle) be measured:

! g(t) !→ g(0)

! g(t) !→!

∞

−∞g(u) du (area)

! g(t) !→!

∞

−∞|g(u)|2 du (energy)

The mathematical term for a measurement is functional.

EE 179, April 4, 2014 Lecture 3, Page 4

Signals and Systems (cont.)

! A system is an object that takes signals as inputs and produces signalsas outputs.

g(t) −→ system −→ f(t)

! In general, the output signal depends on entirety of input signal; e.g.,

! f(t) = d

dtg(t)

! f(t) =! t

t−1g(u) du

! G(s) =!

∞

−∞g(t)e−2πist dt

! Examples of physical systems:

! Electrical circuit: voltage in, voltage or current out

! Building: earth shaking in, building shaking out

! Audio amplifier

EE 179, April 4, 2014 Lecture 3, Page 5

Signal Energy and Power! A signal is periodic if it repeats: g(t+ T ) = g(t) for every t.

E.g., sin t has period 2π and tan t has period π.

! The power of a periodic signal g(t) is

Pg =1

T

"

a+T

a

|g(t)|2 dt

where T is the period of g(t). If g(t) is complex valued, then |g(t)|2 isthe square of magnitude/modulus.

! The power of a general signal is a limit:

Pg = limT→∞

1

T

"

T/2

−T/2

|g(t)|2 dt

This limit may be 0. E.g.,

g1(t) =

#

e−at t > 0

0 t < 0or g2(t) =

1

1 + |t|

EE 179, April 4, 2014 Lecture 3, Page 6

Signal Energy and Power (cont.)

! The energy of a signal g(t) is"

∞

−∞

|g(t)|2 dt

We are interested in energy only when it is finite. Common cases:! Bounded signal of finite duration; e.g., a pulse

! Exponentially decaying signals (output of some linear systems with pulseinput)

! Necessary conditions for finite energy.

! The energy in the “tails” of the signal must approach 0:

limT→∞

$

" T

−∞

|g(t)|2]dt+

"

∞

T

|g(t)|2]dt

%

= 0

! We would expect that the instantaneous power |g(t)|2 → 0, but that isnot required. (This is only of mathematical interest.)

EE 179, April 4, 2014 Lecture 3, Page 7

Units of Power

! If a signal g(t) measured in volts is applied to a load resistor R, then thepower in watts is

P =g(t)2

R

Normally we do not care about the load, so we normalize to R = 1.

! In many applications, the effect of the signal varies as the log of thesignal; e.g., human hearing and sight.

! Power can be expressed in decibels (dB), which are logarithmic andrelative to some standard power. If P is measured in watts, then

! power in dBW is 10 log10 P (power relative to 1W)

! power in dBm (or dBmW) is log10(1000P ) = 30 + 10 log10 P

! One bel (B) is too large to be useful.

The bel is named for Alexander Graham Bell (1847–1922). The dB was adopted by NBSin 1931. It is not an SI unit.

EE 179, April 4, 2014 Lecture 3, Page 8

Classification of Signals

! Signals can have a variety of characteristics, including

! values can be continuous or discrete

! continuous or discrete time variable

! deterministic or random

! For deterministic signals, we have four cases:

! continuous time, continuous valued (mathematics)

! continuous time, discrete valued

! discrete time, continuous valued (digital signal processing)

! discrete time, discrete valued (digital switching)

! Time can be restricted to a finite interval (e.g., periodic)

EE 179, April 4, 2014 Lecture 3, Page 9

Classification of Signals (cont.)

EE 179, April 4, 2014 Lecture 3, Page 10

Operations on Signals

! Time shifting/delay: g(t± T )

EE 179, April 4, 2014 Lecture 3, Page 11

Operations on Signals (cont.)

! Time scaling: g(at) stretches (0 < a < 1) or squeezes (a > 1)

EE 179, April 4, 2014 Lecture 3, Page 12

Operations on Signals (cont.)

! Time reversal: g(−t)

! Each of these operations corresponds to a linear system.

EE 179, April 4, 2014 Lecture 3, Page 13

Unit Impulse Signal! Most physical systems give the same output for any narrow pulse with a

given area.

0 50

5

10

15

0 50

0.5

1

1.5

2

0 50

0.2

0.4

0.6

0.8

0 50

1

2

3

4

5

0 50

0.2

0.4

0.6

0.8

1

0 50

0.2

0.4

0.6

0.8

1

0 50

10

20

30

40

50

0 50

0.2

0.4

0.6

0.8

1

! The abstraction of a infinitely narrow signal with area 1 is the unitimpulse signal. Paul A. M. Dirac “defined” δ(t) by

δ(t) = 0 if t = 0 and

"

∞

−∞

δ(t) dt = 1

! The area of the impulse is important; the energy of δ(t) is not defined.EE 179, April 4, 2014 Lecture 3, Page 14

Properties of Unit Impulse Signal

! Sampling property:"

∞

−∞

ϕ(t)δ(t − T ) dt =

"

∞

−∞

ϕ(t+ T )δ(t) dt

=

"

∞

−∞

ϕ(T )δ(t) dt = ϕ(T )

"

∞

−∞

δ(t) dt = ϕ(T )

In more rigorous mathematics, the sampling property defines the unitimpulse as a generalized function.

! Convolution:

(ϕ ∗ δ)(T ) =

"

∞

−∞

ϕ(t)δ(t − T ) dt = ϕ(T )

! Multiplication by a function:

ϕ(t)δ(t) = ϕ(0)δ(t)

! Fourier transform of unit impulse equals 1 at all frequencies

EE 179, April 4, 2014 Lecture 3, Page 15

Unit Step Function u(t)

! The Heaviside unit step function is defined by

u(t) =

#

1 t > 0

0 t < 0

The unit step function corresponds to turning on at time 0.

! Unit step is integral of unit impulse:

u(t) =

"

t

−∞

δ(u) du ⇒ δ′(t) = u(t)

EE 179, April 4, 2014 Lecture 3, Page 16

Periodic Signals

! A signal g is called periodic if it repeats in time; i.e., for some T > 0,

g(t+ T ) = g(T )

for all t.

! If g is periodic, its period is the smallest such T .

! Examples: trignometric functions are periodic. Period of cos t is 2π;period of tan t is π.

! The period of g(mT ) is T/m. E.g., period of cos 2πft is 2π/f , and itsfrequency is f Hz (cycles per second).

! If g and f are periodic, their common period is LCM(Tg, Tf ). E.g.,period of sinπt+ sin 2πt/5 is LCM(2, 5) = 10.

EE 179, April 4, 2014 Lecture 3, Page 17