continuous-time filtering: modulation and am...

Continuous-time Filtering:Modulation and AM Radio

Ideal filters

• distortion, signal power and bandwidth• filter classifications• impulse responses and causality

Communication systems

• radio transmission system: modulation• demodulation of AM signals• envelope detection of AM signals

Topics

What is distortion? ‘Changing the shape’ of a signal

Multiplication of a signal by a constant (even a negative one)or shifting it in time do not change its shape

No Distortion Distortion

Ideal filters

Filters separate signals in one frequency range from signalsin another frequency range

An ideal filter passes all signal power in its passbandwithout distortion and completely blocks signal poweroutside its passband

Ideal filters

Filter Classifications

FT analysis

The Fourier Transform of a signal is

For real-valued signals:

Fourier Transforms provide a different perspectiveon a signal. Some aspects become more evident inthe frequency domain.

!

f (t) =1

2"F( j#)e j#t

d#$%

+%

& ' ( ) F( j#) = f (t)e$ j#t

dt$%

+%

&

!

f (t) =1

"|F( j#) | cos(#t +$F( j# ))dt

0

+%

&

Signal energy

The signal energy can be computed in the time domain or inthe frequency domain.

This is Parseval’s Theorem:

Most of the energy of physical signals is concentrated in parts ofthe spectrum.

Example: Humans cannot hear above 20 KHz. Hence the usefulpart of audio signals have frequencies below 20 KHz. Wesay that audio signals are low-pass signals and have abandwidth of 20 KHz.

!

f (t)2dt =

"#

+#

$1

2%F( j&)

2d&

"#

+#

$

Signal bandwidth

Bandwidth means “range of frequencies.” We use it denote rangewhere most of signal energy is concentrated

E.g., for a low-pass signal, frequency such that

For r = 0.99, we say that is the 99% bandwidth of the signal.

For band-pass signal, frequency range such that

For r = 0.99 then we say that is the 99% bandwidth ofthe signal.

!

1

2"F( j#)

2d#

$%

+%

& = r1

2"F( j#)

2d#

$'

+'

&

!

"

!

"

!

1

"F( j# )

2d#

+$ l

+$u

% = r1

2"F( j# )

2d#

&'

+'

%

!

" ="u#"

l

Why are ideal filters called ideal?

One reason is that perfect circuit components with ideal characteristicsdo not exist

Another, more fundamental, reason can be seen in the correspondingimpulse responses

All responses have nonzero response before the impulse is applied at

!

t = 0

Ideal filters are non-causal

For example, ideal lowpass filter has a frequency response

The inverse Fourier transform is the impulse system response

!

H( f ) = A rect( f /2 fm )e" j2#ft0

!

h(t) = 2Afmsinc(2 fm (t " t0))

Non-causal system!

Ideal filters cannot bephysically realized, butthey can be closelyapproximated

Ideal filters




Topics

What is modulation?

Modulation is the process of shifting the frequency of a signalso that the resulting signal is in a desired frequency band

Why would one be interested in modulation?1. Antenna length requirements: for efficient radiation, antenna

length is 1/2 or 1/4 of the wavelength to be radiated

Here, c is the speed of light. At a frequency of 1 KHz (humanvoice), a 1/4 wavelength antenna would measure

However, had the signal higher frequency, smaller antennaswould do

!

"c =c

fc

!

L =c

4 fc=3 "10

8

4 "103

=3

410

5= 75km

Why modulation?

Why would one be interested in modulation?2. Interference: two communications, at the same time and

over the same geographical area, using the samefrequency, would interfere with each other.

This can be solved by shifting information signals todifferent frequencies

Frequency band Designation Typical users

3-30 kHz Very low frequency (VLF) Long-range navigation

30-300 kHz Low frequency (LF) Marine communications

300-3000kHz Medium frequency (MF) AM radio broadcasts

3-30 MHz High frequency (HF) Amateur radio; telephone

30-300 MHz Very high frequency (VHF) VHF TV; FM radio

0.3-3 GHz Ultrahigh frequency (UHF) UHF TV; radar

3-30 GHz Superhigh frequency (SHF) Satellite communications

FCC Frequency Band Assignments

Modulation and Radio

A simplified radio communication system can be the following:

Music, people talking produces the input signal

In the transmitter, the signal is modulated, here as an amplitudemodulated (AM) signal

The frequency is the carrier frequency of the radio station.

Usually , where is the bandwidth of

!

x(t " t0)

!

x(t)

!

cos("ct)

!

cos("c(t # t

0))

!

x(t)

!

x(t)cos("ct)

!

"c

!

"c# $

!

"

!

"

!

"

!

x(t)

Modulation and Radio

A simplified radio communication system can be the following:

The radio signal is broadcast and picked up by the antenna on thereceiver.

The receiver converts down the broadcast signal (demodulates) intoa signal proportional to with a small time delay (lessthan a millisecond)

!

x(t " t0)

!

x(t)

!

cos("ct)

!

cos("c(t # t

0))

!

x(t " t0)

!

t0

!

"

!

"

Amplitude Modulation in the Time Domain

Input signal:

AM modulated signal:

!

x(t)

!

x(t)cos("ct)

Amplitude Modulation in the Frequency Domain

Input signal:

AM modulated signal:

!

x(t)" # $ X( f )

!

x(t)cos(2"fct)# $ % Y ( f ) = X( f ) * 1

2&( f ' fc ) + 1

2&( f + fc )( ) = 1

2X f ' fc( ) + 1

2X f + fc( )

Modulation and Fourier Transform

Different AM stations have different carrier frequenciesIn the US, the carrier frequency is chosen from a set spaced10 KHz apart. These are taken within the frequency rangethat starts at 540 KHz and ends at 1700 KHz.

With a 10 KHz operation bandwidth, each AM station canbroadcast a signal of 5 KHz bandwidth.

Comparison with Frequency Modulation (FM)

Comparison of a signal and an FM version of it:

FM transforms a signal into another with higher-frequency bandbut unit magnitude

Stations range from 88 MHz to 108MHz at 200 KHz apartBandwidth of each station is only 15 KHzFM Antenna is only

!

L =c

4 fc=3 "10

8

4 "108

=3

4= 0.75m

!

cos "c

+ #x(t)( )t( )

!

x(t)

Other types of Modulation

In general, modulation is a process that changes one or moreproperties of a periodic signal (its amplitude, its frequency orits phase) by another signal

In the previous examples, the modulated, periodic signal is acos function, while the modulating signal is the radio wavesignal that we would like to transmit

Instead of cos, other periodic signals can be used. For example,using a pulse wave instead of a cosine wave results intopulse modulation.

Pulse modulation is used to transfer analog signals as digitalsignals (that is, as a sampled and quantized signal) by usinga digital transmission system

Ideal filters




Topics

Coherent Demodulation of AM signals

An antenna for an AM radio receiver captures signals from tensof radio stations. How does it select the right signal?

The receiver uses a band-pass filter with an adjustable pass-band to tune in

The band-pass output is then converted into an audio signalthat is proportional to . Here, is the time delay ofthe propagation channel

!

f (t)

!

cos("ct)

!

cos("c(t # t

0))

!

f (t " t0)

!

f (t)cos("ct)

!

f (t " t0)

!

t0

!

"

!

"

Coherent demodulation of AM signals

The receiver input will be a signal

The receiver output will be a signal

The Fourier Transform of m(t) is

The first term of M(jw) is the signal we would like to recover

!

r(t) = kf (t " t0)cos(#c (t " t0))

!

M( j") =k

2F( j")e# j"t0 +

k

4F j " # 2"c( )( ) + F j " + 2"c( )( ){ }e# j"t0!

m(t) = r(t)cos "c (t # t0)( )

=k

2f (t # t

0){1+ cos(2"c (t # t0))}


Graphically, we have

In the final step,we take

!

r(t)" R(#)!

f (t)" F(#)

!

m(t)"M(#)

!

|F(") |

!

|R(") |

!

HLPF(")

!

|M(") |

!

"#c

!

"#c

!

"#c

!

"#c

!

"2#c

!

"2#c

!

"2#c

!

"2#c

!

2"c

!

2"c

!

2"c

!

2"c

!

"c

!

"c

!

"c

!

"c

!

1

!

k /2!

k /2

!

1

!

Y (") = HLPF(")M(")


After taking , we obtain

What happens if the receiver multiplies the broadcast signal

By with ?

For optimal demodulation the receiver must know exactly thevalue , hence the term coherent demodulation.

To avoid the estimation of an accurate , another processcalled envelope detection is used in practice

!

Y (") = HLPF(")M(")

!

y(t)"Y (#)

!

y(t) =k

2f (t " t

0)

!

t0

!

t0

!

r(t) = kf (t " t0)cos(# c (t " t0))

! !

t1" t

0

!

cos("c(t # t

1))

!

m(t) = r(t)cos " c (t # t1)( )

=k

2f (t # t

0){cos(" c (t0 # t1)) + cos(2" c (t # t0 + t

1))}

Ideal filters




Topics

Recall AM Signal in Time Domain

Input Signal: AM Modulated Signal:

Envelope of signal is proportional to

!

f (t)

!

f (t)cos("ct)

!

f (t)

Recall AM Signal in Time Domain

Input Signal: AM Modulated Signal:

Envelope of signal is proportional to

!

( f (t) +")cos(#ct)

!

f (t) +" =| f (t) +" |

!

" >max | f (t) |

!

f (t) +" =| f (t) +" |

Envelope detection of AM signals

In order to implement envelope detection at the AM receiver,the AM transmitter is modified as follows

!

f (t)

!

"

!

cos("ct)

!

( f (t) +")cos(#ct)

!

+

!

"

!

" >max | f (t) |


The AM receiver will extract the signal as follows

!

r(t) = ( f (t) +")cos(#ct)

!

p(t) =| r(t) |

!

p(t)

!

q(t) = f (t) +"

!

HLPF(")

!

q(t) = f (t) +"


To see how the signal extraction works, let us look atthe signal/system interaction in the FD

First, is a periodic signal of frequency ,thus it can be expanded as

In this way, can beexpressed as

!

| cos("ct) |

!

2"c

!

| cos("ct) |=

a0

2+ a

ncos(n2"

ct)

n=1

+#

$

!

p(t) =| r(t) |= ( f (t) +") | cos(#ct) |

!

p(t) = p1(t) + p

2(t)

!

p1(t) = f (t)cos("ct) =

a0

2f (t) + an f (t)cos(n2"ct)

n=1

+#

$

!

p2(t) =" cos(#ct) =

a0

2" + an" cos(n2#ct)

n=1

+$

%


Now, consider that is designed in such a way that

Then, the response of to is justTo determine the filter response to , first we obtain

Because only the first term is within the pass-band of the filterwe just obtain

Therefore, using superposition, we find the output of

!

HLPF(")

!

p2(t)

!

HLPF(0) =

2

a0

!

HLPF(n2"

0) = 0

!

n "1

!

HLPF(")

!

q2(t) ="

!

p1(t)

!

P1(") =

a0

2F(") +

an

2{F(" # n2"

c) + F(" + n2"

c)}

n=1

+$

%

!

HLPF(")

!

Q1(") =

a0

2H(0)F(")# q

1(t) = f (t)

!

q(t) = q1(t) + q

2(t) = f (t) +"

!

HLPF


!

f (t)" F(#)

!

r(t)" R(#)

!

p(t)" P(#)

!

Q(") = HLPF(")P(")

!

HLPF(")

Graphically, we have

In the final step,we take

!

|F(") |

!

|R(") |

!

|P(") |

!

"#c

!

"2#c

!

2"c

!

"c

!

"#c

!

"2#c

!

2"c

!

"c

!

"#c

!

"2#c

!

2"c

!

"c

!

"#c

!

"2#c

!

2"c

!

"c


The process of envelope detection is insensitive totime shifts in the carrier wave

In other words, if the input to the detector is

The detector output will still be becausethe coefficients of the CTFS of donot change with a shift in time

!

r(t) = ( f (t " t0) +#)cos($c (t " t0))

!

f (t " t0) +#

!

| cos("c(t # t0)) |

Modulation and AM Radio Summary

Radio and Communication Systems can be better understood inthe Frequency Domain and by looking at signals spectra:

Before transmitting, signals need to be transformed from low-pass to band-pass signals.

-This is done by; e.g. AM modulation: or

At the receiver, signals are detected using demodulation and alow-pass filter. We discussed here two approaches:

- Coherent detection

- Envelope detection

!

f (t)

!

( f (t) +")cos(#ct)

!

f (t)cos("ct)

!

m(t) = r(t)cos("c(t # t

0))

!

r(t)

!

HLPF(")

!

r(t)

!

p(t) =| r(t) |

!

HLPF(")

!

k

2f (t " t

0)

!

f (t " t0) +#

Applications to radio-controlled devices

At its core, we have seen that radio is a very simple technology,yet the impact of radio in our society is tremendous. Some ofthe technologies that depend on radio include the following:

AM and FM radioCordless phonesGarage door openersTV remote controlWireless networksRadio-controlled toysGPS receiversSatellite communications…

Note that radio waves not only can be used to transmitinformation, but also that this information can be used forremote control


A simplified block diagram of a remotely controlled toy is thefollowing:

By turning a knob in the joystick, an electric circuit is closed. Thiscircuit is connected to an Integrated Circuit which generates thesignal to be transmitted. It is possible to generate different signals

Before transmission, the signal is pulsed-modulated, to place the signalfrequencies in a range anywhere between to

!

27MHz

!

49MHz


At the other end, the truck receiver is constantly monitoring forincoming signals

The picked signals are passed through a band-pass filter to block outthe frequencies outside the chosen frequency range

The obtained pulse sequence is then sent to an Integrated Circuit inthe truck. The IC decodes the sequence to generate the rightcontrol for the motor of the truck!

continuous-time filtering: modulation and am...

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