Signal Transmission Through LTI SystemsEE 442 Lecture 3

1Signal Transmission

The defining properties of any LTI system are linearity and time invariance.

https://en.wikipedia.org/wiki/Linear_time-invariant_system

From EE 400: Convolution

s j=

Signal Transmission 2

Information converted into an electrical waveform suitable for transmission is called a signal. Signals are time-varying and may be either analog or digital representations.

A signal is a waveform representing or encoding information or a message. It may be a voltage, current, electromagnetic field or other physical parameter.

A signal may be either analog or digital. Sometimes the terminology of continuous or discrete is used to distinguish between analog and digital signals.

Almost all practical waveforms can be analyzed and constructed from many harmonically-related sinusoidal waveforms. If they are periodic, repetitive waveforms they may be expressed as a Fourier series,

It is valuable to be able to interpret a signal in both the time domain and the frequency domain.See next page for figure.

Summary of Lecture 2 – Page 1

( )0 0 0

1

( ) cos ( ) sin ( )n n

n

f t a a n b n

=

= + +

0 0 0

0 0 0

1 2 2( ) , ( ) cos( ) , ( )sin( )

T T T

n na f t dt a f t n t dt b f t n t dtT T T

= = =

Signal Transmission 3

Phasors are convenient for representing sinusoidal signals and written as

The Euler identity is used in relating sinusoidal signals to phasors,

The phasor projection (phasor has counterclockwise rotation in complex plane) upon the real axis is the real cosine waveform and upon the imaginary axis is the real sine waveform in time.

Negative frequencies are a mathematical construct to analyze real signals using the complex number framework. It requires the use of double-sided spectra.

For an analog signal the time-varying parameter of the signal represents a physical parameter that takes on a continuous value within a defined range. For digital signals, its time-varying values take on discrete values from an assigned set of values or states (a unique symbol is assigned to each).

Summary of Lecture 2 – Page 2

j cc c e=

= cos( ) sin( )j te t j t

Signal Transmission 4

Signals are usually categorized as (1) analog versus digital, and (2) continuous versus discrete.

Digital communication involves a bit sequence (or symbol sequence) that are not periodic (or predictable). Periodic waveforms (such as carrier signals and clock signals) do not convey data or information.

Summary of Lecture 2 – Page 3

Signal Transmission 5

Steady-State Response in Linear Time Invariant (LTI) Network

By steady-state we imply a sinusoidal excitation and response.

LTI Networkh(t) and H(f)

A sinusoidal signal of frequency f at the input, x(t), produces a sinusoidal signal of frequency f at the output, y(t). The output y(t)is given by

x(t) y(t)

H(f) will modify input x(f) by a change in magnitude and phase only.

The frequency f is unchanged and the output is causal.

X(f) Y(f)

( ) ( ) ( )=Y f H f X f

6Signal Transmission

Pulse Response in Linear Time Invariant Network

We are interested in the pulse response of a given LTI system with abounded input – bounded output (BIBO).

LTI Networkh(t) & H(f)

( )

( )

x t

X f

( ) ( ) ( )

( ) ( ) ( )

y t x t h t

Y f X f H f

=

=

where x(t) is the input, h(t) is impulse response of the network and

y(t) is the output (Note: the symbol denotes convolution).

+

=

=

( )

( ) ( ( ) ( ))

( ) ( ), ( ) ( ) ( ) ( )

where ( ) is the transfer function of the network.

We can write ( ) ( )

and ( ) ( ) ( )

h

y x h

j f

j f j f f

x t X f h t H f and y t Y f

H f

H f H f

Y f X f H f

e

e e

by convolution theorem

convolution

Signal Transmission 7

Convolution Theorem (This should be review for you)

The convolution theorem states that under suitable conditions the

Fourier transform of a convolution operation is the product of

Fourier transforms.

Convolution:

InverseFourier

Transform

FourierTransform

FourierTransform

Signal Transmission 8

Example: Impulse Response of a LTI Network

( )h t( )x t

This is the transient response of a Linear Time Invariant (LTI) RC network.

Unit impulseresponse h(t)( ) ( )x t t

LPF

Signal Transmission 9

Signal Distortion During Signal Transmission

In linear amplifiers and channel transmission we want the output waveform to be a replica of the input waveform. That means we want distortionless transmission.

In other words: If x(t) is the input signal, the output signal y(t) is required to be of the form,

y(t) = K· x(t - td), (where K is a constant)

Thus, y(t) has its amplitude modified by factor K and is time delayed by time td.

And in the frequency domain,

The Fourier transform is Y(f ) = K·X(f )e-j2 f td

by direct application of the convolution theorem.

10Signal Transmission

Signal Distortion During Signal Transmission

For distortionless transmission the transfer function H(f ) is of the form,

−=

= − = −

2From ( ) ( )

and ( ) 2h d d

dj ftH f H f

f f t t

e

Conclusion: Distortionless transmission requires a constant amplitude|H(f)| over frequency and a linear phase response h(f) passing through the origin at f = 0.

|H(f)|

h(f)

f

A

11Signal Transmission

All-Pass Systems versus Distortionless Systems

All-Pass System: Has a constant amplitude response, but doesn’t alwayshave a linear phase response.

A distortionless system is always an all-pass system, but the converse isnot true.

The transmission phase characteristic, if it doesn’t have a constant slope, causes distortion in its output.

All practical LTI systems only approximate a linear phase response passingthrough f = 0. Hence, from the time delay expression,

or the time delay td is frequency dependent.

For EE 442: Phase distortion is important in digital communication systems because a nonlinear phase characteristic in a channel causespulse dispersion (i.e., results in spreading) and causes pulses to interferewith adjacent pulses (called Inter-Symbol Interference).

= −

( )( ) either has a constant slope, hd

ft f

Signal Transmission 12

http://www.slideshare.net/simenli/ch1-2-49340691

Waveguide

PC board substrate

Signal Transmission 13

Time Delay of Ideal Transmission Line

What is the time delay of a coaxial transmission line of length L?

The delay varies linearly with the length of the transmission line.

ZL = ZS = 50

L

2 cycles → 4 radiansdelay shown

(radians)

(radians/sec)

and

d

d

L

velocity

t

t

= −

=

1x

direction of travel

0

Phase delay:

Transmission Line

ZL

ZS

VS

Signal Transmission 14

Phase Delay in an Ideal Transmission Line

ZL = ZS = 50

L

50 Transmission Line

ZL

ZS

VS

For an air-filled coaxial transmission line the time delay is roughly 1 nanosecond (10-9 second) per foot of physical length L. For a transmission line with a polyethylene dielectric the time delay isof the order of 1.5 nanoseconds per foot of length.

2 f =

= −dt

Length = L

Linear Phase

Response

15Signal Transmission

Example: Lumped Component RC Network (Low-Pass Filter)

R

Cg(t) y(t)

+ +outputinput

We have an RC low-pass filter, find the transfer function H(f), and sketch|H(f)|, the phase n(f) and delay (or group delay) td(f).

The transfer function H(f) is given by y(f)/g(f),

( )( )

= =

++

11

( ) general form: where1 (2 )(2 )

aRCH f aa j f RCj f

RC

1RC

a = =

16Signal Transmission

RC Network LPF Example (continued)

( )( )

( )

( ) ( )

( )

( ) ( )

− −

= =

++

=

+

= − = −

− = −

2 2

1 1

1

1

1( ) general form: where

(2 )(2 )

Therefore, the magnitude ( ) and phase ( ) are given by

1( ) , and

1 2

2( ) tan tan 2

1( ) 2 2

h

h

h

RC

RC

aH f a

a j f RCj f

H f f

RCH f

fRC

ff f RC

a

f f RC RC for fRC

Magnitude:

Phase:

Low frequencyapproximation

17Signal Transmission

RC Network LPF Example (continued)

The delay td = negative derivative of the phase with respect tofrequency, therefore, (often called the group delay)

1

2

(tan ) 1From a table of derivatives:

1

d x

dx x

−

=+

( )

−

− = − = −

= − = − −

1

1

( ) ( ) ( )1( ) , By definition

(2 ) 2

is the slope of the phase versus frequency.

For our RC low-pass filter example the group delay is

( ) 2( ) tan

(2 ) (2 )

h h hd

hd

RC

d f d f d ft f

d d f df

d f fdt f

d f d f( )( )

−

= − −

1tand

RCd

Signal Transmission 18

( )( )

( )

( ) ( )

( )

( )

( )

−− = − −

−= − =

+ +

= =++

= = = =

1

2

2 22

2 22

( )( ) tan 2

(2 ) (2 )

121( )

(2 )1 2 1 (2 )

1( )

1 ( )1 ( )

Of course, for very low frequencies:

1 1 1( ) for 2

hd

d

d

d

d f dt f f RC

d f d f

d f RC RCt f RCd ff RC f

RC

RCRCt fRC

RC

t f RC f aa RC

RC Network LPF Example (continued)

19Signal Transmission

Slope = - td

Amplitude response within 2% of peak value

(constant phaseshift asymptote)

Amplitude response is 0.707 of peak value

RC Network LPF Example (continued)

( )h

0

( )1( ) tanh RC −= −

From Lathi & Ding, 4th ed., 2008; Figure 3.28 (p. 129)

20Signal Transmission

RC Network LPF Example (continued)

From Lathi & Ding, 4th ed., 2008; Figure 3.28 (p. 129)

− ( )dt f

1

a

2 f =a0

= = =3

1 1dBRC and

a RC

2%

− = − = −

( ) ( ) ( )1( )

(2 ) 2h h h

d

d f d f d ft f

d d f df

3dB

21Signal Transmission

An Ideal Filter with an Infinitely Sharp Cutoff

The signal g(t) is transmitted without distortion but has a time delay td.

( )1

2

2

( ) and ( ) 2 ;2

( ) and2

( ) 2 sinc 2 ( )2

h d

d

d

d

j ft

j ft

fH f f ft

B

fH f

B

fh t F B B t t

B

e

e

−

−

−

= = −

=

= = −

dtBB−

( )H f ( )h t

( ) 2h df ft = −

1

2B

CAUSALITY VIOLATION

f

t

1 “Brick Wall Filter” Unit impulseresponse h(t)

From Lathi & Ding, 4th ed., 2008; Figure 3.29 (p. 130)

22Signal Transmission

Ideal Filters versus Practical Filters

Impulse response h(t) is the response to impulse (t) applied at time t = 0.

The ideal filter on the previous slide is noncausal → unrealizable .

The practical approach to filter design is for h(t) = 0 for t < 0.

A good approximation if td is large. Many different practical “non-ideal” filters exist, for example:

ˆ( ) ( ) ( )h t h t u t=

Butterworth Chebyshev I

Chebyshev II Elliptic

Signal Transmission 23

Nth order

Butterworth Filter (Maximally Flat Magnitude)

(nth order filter)

https://www.youtube.com/watch?v=dmzikG1jZpU

of

n sections

Linear Logarithmic

Signal Transmission 24

Butterworth Filter (Maximally Flat Magnitude)

Sallen–Key topology

“Cauer topology”

22

2

(0)( )

1

n

C

HH j

=

+

Unit circle

Re(s)

Im(s)

4th order Butterworth Low-Pass Filters j = +

Pole-zero diagram

n = 3

Signal Transmission 25

Butterworth Filter Impulse Response

( )h t

( )H f

( )h f

Fourth-order filter (n = 4)

After Lathi and Ding, 4th ed. 2009; Figure 3.33 (p. 133)

Signal Transmission 26

Comparing Butterworth, Chebyshev & Bessel Filters

http://www.analog.com/library/analogDialogue/archives/43-09/EDCh%208%20filter.pdf?doc=ADA4666-2.pdf

h(t)

Unit step response

|H(f)||H(f)|

Unit impulse response

Signal Transmission 27

Bessel-Thompson Filter Characteristics

Group Delay Flatness

https://www.sciencedirect.com/topics/engineering/bessel-filter

In signal processing, a Bessel filter is a type of analog linear filter with a

maximally flat group/phase delay (maximally linear phase response),

which preserves the wave shape of filtered signals in the passband.

Signal Transmission 28

Phase Delay versus Group Delay

0

0

0

0

0

at one frequency

frequency

Phase response of ( ) is ( ), therefore

( )Phase delay ( ) is ,

2

( )Group delay (at one )

(2 )

ph

gr

f

ff

f

H f f

fand

f

d

d

= −

= −

Input:

( ) ( ) cos(2 )

Output:

( ) ( ) ( ) cos(2 ( ))grp ph

g t A t ft

y t H f A t t f t t

=

= − −

Envelope

Example:

Signal Transmission 29

Phase Delay versus Group Delay

|H(f)|

(f)

f

A

In this case the phase delay is equal to the group delay.

0

0

0

0

0

at one frequency

frequency

( )Phase delay ( ) is , and

2

( )Group delay (at one )

(2 )

ph

gr

f

ff

f

f

d f

d

= −

= −

0f

Signal Transmission 30

Phase Delay versus Group Delay

|H(f)|

(f)

f

A

0f

0( )f−

0( )

( )gr

f

d f

d

= −

0

0

( ) ph

f

f

= −

Linear phasecharacteristic

Signal Transmission 31

Group Delay (or Envelope Delay)

Group delay gr is a valuable way to describe a filter's pass-band behavior.

Simple example: Consider a square wave, which is composed of many frequency components. A square wave remains square only if its frequency components stay in proper phase alignment with one another. If we pass the square wave through a network and expect it to remain square, then we need to ensure that the network doesn't misalign the phases of each of these frequency components.

Group delay is:

(1) A measure of a network’s phase distortion.(2) The transit time of a signal’s envelope through a device versus frequency.(3) The derivative of the device's phase characteristic with respect to frequency (i.e., mathematical interpretation).

Signal Transmission 32

Group Delay (continued)

Another viewpoint on Group Delay in networks:

Phase is never constant over a frequency band and each frequency doesn’t take the same amount of time to travel through a network, or over a channel. Thus, Group Delay variation causes distortion of the signal waveform as it passes through a network, or travels over a channel.

Group Delay characterizes deviations from the ideal, linear Phase Delay over all frequencies.

Signal Transmission 33

Amplitude and Phase Response for a Bandpass Filter

Ideal linear phasehttps://fr.wikipedia.org/wiki/Retard_de_groupe_et_retard_de_phase

Actual phase

Signal Transmission 34

Group Delay (continued)

https://www2.units.it/ramponi/teaching/DSP/materiale/Ch4%282%29.pdf

Input

Output

ph gr Group delay of the envelopePhase delay

carrier

envelope

Signal Transmission 35

Summary of Channel Impairments (Overview)

• Linear distortion (caused by impulse response)

H(f) attenuates and phase shifts the signal.

• Nonlinear distortion (such as “clipping”)

• Random Noise (independent additive random signals)

• Interference (from other transmissions or sources)

• Self interference & ISI (from reflections and multipath)

( ) ( ) ( ) ( ) ( ) ( )y t h t g t Y f H f G f= =

−

( )

p

p

y

y t

y

−

−

( )

( )

( )

p

p p

p

y t y

y y t y

y t y

( )y t =

ISI is intersymbol interference

Signal Transmission 36

Pulse Distortion in a Sine-Squared Shaped Pulse

Input pulse

Input pulseInput pulse

Output pulse

Output pulse

Output pulse

(a) Amplitude-frequency distortion and phase-frequency distortion combined

(b) Amplitude-frequency distortion (c) Phase-frequency distortion

http://users.tpg.com.au/users/ldbutler/Measurement_Distortion.htm

This is typically whatpulse distortion looks like in channels.

Signal Transmission 37

Wireless Transmission Channels

EE 442 Lecture 3continued

Signal Transmission 38

The Advent of Radio – Some History

British Post Office engineers inspect Guglielmo Marconi's wireless

telegraphy (radio) equipment, during a demonstration on Flat Holm

Island, May 13, 1897. This was the world's first demonstration of

the transmission of radio signals over the open sea, a distance

spanning about 3 miles.

https://www.allatsea.net/vhf-radio-distress-call/

Signal Transmission 39

Question: Why is wireless communication so challenging?

https://www.semanticscholar.org/paper/Evolution-toward-5G-multi-tier-cellular-wireless-An-Hossain-Rasti/bdd9347c8a98be34f6c7fee29b59fb5a6f800c71

Actually this is a preview of 5G cellular.

Signal Transmission 40

Wireless Channel: Atmospheric Radio Wave Propagation

https://physics.stackexchange.com/questions/192625/is-the-wobbly-rope-depiction-of-a-radio-wave-inherently-wrong-and-how-do-vector

Electric field

Magnetic field

Is this transmission channel analogous to a linear time-invariant network?

Signal Transmission 41

This is what limits the range.

Free Space Path Loss (FSPL)

https://en.wikipedia.org/wiki/Free-space_path_loss

( ) ( )

( ) ( )

10

10

10 10 10

10 10

22

2

44FSPL

where

4FSPL(dB) 10 log

4FSPL(dB) 20 log

4FSPL(dB) 20 log 20 log 20 log

FSPL(dB) 20 log 20 log 147.55 dB

rfr

c

c f

rf

c

rf

c

r fc

r f

= =

=

=

=

= + +

= + −

No reflections,diffraction

or scattering

r

Signal Transmission 42

( )2

4rt tr G G

rP P

=

ReceivedPower

TransmittedPower

Antenna Gains Separation

Distance

Wavelength

Antenna-to-Antenna Path Loss – The Friis Formula

TransmittingAntenna(Gain Gt )

ReceivingAntenna(Gain Gr )r

rP

tP

No reflections,diffraction,

or scattering.

Signal Transmission 43

Antenna-to-Antenna Path Loss – The Friis Formula

TransmittingAntenna(Gain Gt )

ReceivingAntenna(Gain Gr )r

rP

tP

No reflections,diffraction,

or scattering.

( )2

4rt tr

cG G

rfP P

=

ReceivedPower

TransmittedPower

Antenna Gains Separation

Distance

Frequency

Speed of Light

Signal Transmission 44

Types of Radio Wave Propagation in Atmosphere

Earth

Earth

Ionosphere

Earth

Sky-wave

Line-of-sight

Ground-wave

SatelliteSpace-wave

• Space-wave propagation

Satellite to ground and vice versa

• Sky-wave propagation– Limited to 2 to 30 megahertz (MHz) – Long-distance coverage is due to

ionosphere reflection (U-shape due to variation in index of refraction)

– Used in international broadcasting, amateur radio, & military communication

• Line of Sight (LOS) propagation– Above 30 MHz (SHF band)– EM propagates in straight line (LOS)– FM radio & TV analog broadcasting

• Ground-wave propagation– Below 2 MHz (LF band)– EM wave follows the Earth’s contour

due to diffraction– Applies to AM radio broadcasting

From EE101A Lecture (D. B. Estreich – Spring 2015)

Signal Transmission 45

Wireless Channel Wave Propagation

https://slideplayer.com/slide/12737878/

Signal Transmission 46

Causes of Deterioration of Wireless Signals

1. Free-space loss (scattering, molecular absorption, fog or rain, and wavefront spreading) – large-scale fading

2. Multipath (scattering from objects: reflection, diffraction and refraction) – small-scale fading

3. Shadowing – large-scale fading 4. Mobility (Doppler shifting, rapid environment changes from moving)5. Interference (from other communication systems)6. Noise pickup from the channel (also transmitters and receivers

add noise) – generally AWGN

Added Complication: generally the channel is always changing over time.

Signal Transmission 47

Categories of Large-Scale and Small-Scale Fading

Fading

Large-Scale Fading Small-Scale Fading

Path Loss(dB/distance)

Shadow Loss(varies aroundthe path loss)

Multi-PathRayleigh Fading(varies /2)

Signal Transmission 48

http://www.informit.com/articles/article.aspx?p=2832588&seqNum=5

Behavior of large-scale (distance-dependent path loss and

shadowing) and small-scale (fading) propagation effects.

Behavior of Large-Scale and Small-Scale Propagation

Signal Transmission 49

Simplified Path-Loss Model for Wireless

d0 = reference distance from antenna to onset of the far-field region(often 1 to 10 meters indoors and 10 to 100 meters outdoors), and

K = “path-loss factor” constant (consists of antenna and average channel

attenuation). If unknown, then use

= “path-loss exponent” (empirically determined)

0tr

dP PK

D

=

0

2

4K

d

=

Environment Path-loss exponent

Free space 2

Urban area cellular radio 2.7 to 3.5

Shadowed area cellular radio 3.0 to 5.0

In-building line of sight 1.6 to 1.8

Obstructed path in building 4.0 to 6.0

Signal Transmission 50

Present Wireless Communication Challenges

Today’s biggest wireless challenges:

1. Scarcity of radio spectrum (limited “sweet spot”)2. Ever increasing data rates are required (think 5G)3. Multitude of different environments factors:

scattering, shadowing, reflection, diffraction, fading, etc.

4. Power consumption in portable user equipment5. Software complexity of to support user mobility6. Cost of infrastructure

→ All of this impacts the challenges of 5G Cellular.

Signal Transmission 51

Questions?

What is going on in this diagram?

Signal Transmission 52

Auxiliary Slides

Signal Transmission 53

https://www.electronics-tutorials.ws/accircuits/series-resonance.html

https://www.electronics-tutorials.ws/accircuits/parallel-resonance.html

0

0

02

1 1( )

2 ( )

( ) R

S

R RQ RC

L LC

and Y jRC RH j

IH j

I

= = =

= =

000

2

( )2 ( )

( ) R

m

L LCQ RC

R R

R Rand Z j

L H j

VH j

V

= = =

= =

RV+ −

Reference: Charles A. Desoer and Ernest S. Kuh, Basic Circuit Theory, McGraw-Hill Book Company, 1969; page 316.

mV+ −

SI

RI

( )Y j

RLC Resonant Circuits (Parallel & Series)

Signal Transmission 54

RLC Resonant Circuits (Parallel & Series)

2 00

0

0

1 1( ) ;

21

H j where QLC

j

= =

+ −

If Q > ½ (underdamped), the natural frequencies are

2 220 0

1; 14d djQ

− = − = −

If Q >> 1, 0d

Signal Transmission 55

Universal Resonance Curve

http://elektroarsenal.net/band-pass-filters-for-if-amplifiers.html

3-dB angular cutoff frequency:

1 0 0

2 0 0

11

2

11

2

Q

Q

− = −

+ = +

3-dB Bandwidth:

02 1 2

Q

= − =

Signal Transmission 56

Phasor Representation of RL and RC Circuits

Signal Transmission 57

Overview of phase shifts and delays

Parameter Expression or Equation Units

Phase shift |H()| radians

Phase delay tph seconds

Group delay td secondsd

dt

d

= −

pht

= −

N+

-

+

-

vin vout+

-

vin

+

-

vout

Signal Transmission 58

Low-Pass filter Transfer Function Calculations

( )

( )

2 2 2 2 2 2

2 2 21 1 1

2 2 2

1

1 (1 )( )

(1 ) (1 )

1( ) Re ( ) Im ( )

(1 ) (1 )

The angle is given by

Im ( ) (1 )( ) tan tan tan

1Re ( )

(1 )

( ) tan

Bec

j RCH f

j RC j RC

RCH f j H f j H f

R C R C

RC

H f R Cf RC

H f

R C

f RC

− − −

−

−=

+ −

−= + = +

− −

− − = = = − −

= −

1 1ause we know that tan ( ) tan ( )z z− −− = −

R

Cvin(t) vout(t)

+ +outputinput