ELEC3540 Lecture 2 Analog Baseband Transmission

ELEC3540 Analog and Digital Communications

ELEC3540-2012-T2-PSB ACADEMY-Lecture 2 1

ELEC3540 Analog and Digital Communications Lecture 2: Analog Baseband Transmission

• Analog baseband

• Discrete baseband

• Performance and Reception


SHANNON’S CHANNEL CAPACITY THEOREM

• Consider the receiver as a combined demodulator/decoder

• Si and Ni are the average signal and noise powers for the input

• So and No are the average signal and noise powers for the output

• BT is the transmission bandwidth and Bm is the message bandwidth

• In general (but not always), a practical receiver improves the signal-to-noise ratio (SNR) in exchange for signal bandwidth



SHANNON’S CHANNEL CAPACITY THEOREM

• By choosing a modulation/coding scheme such that BT >> Bm it should be possible for the receiver to deliver high-quality signal despite a low Si /Ni

• Coded and uncoded modulation schemes permit this type of exchange: an uncoded scheme that does this is FM

• Shannon’s Capacity is given as

where C = channel capacity, B = channel bandwidth (in Hz),

S and N are the average received signal and noise powers respectively

• The noise is additive white Gaussian noise • The equation is used for various practical modulation

schemes since it defines the absolute maximum information rate, R (< C), which can be reliably (without error) sent over the channel

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SC Blog 1

N


ANALOG BASEBAND

Analog baseband – analog signals whose Fourier Transform occupy frequencies extended to (or close to) 0 Hz

Baseband – frequencies are NOT shifted to non-zero point on the frequency axis

Analog baseband signals occupy relatively low frequencies and not suited for transmission through bandpass channels

They are usually transmitted through cables or wires, e.g. telephony



ANALOG BASEBAND

Human ears respond well to signals from about 20 Hz to 20 kHz

The upper frequency cutoff decreases with age, hearing aids help by amplifying high frequency components

Human speech is between 300 Hz to 3.3 kHz, where most energy is

Music covers higher than 3.3 kHz, up to more than 15 kHz

AM radio transmits frequencies up to 5 kHz

FM radio transmits frequencies up to 15 kHz

ELEC3540-2012-T2-PSB ACADEMY-Lecture 2

• Discrete signal = analog signal being represented as a list of numbers

• These numbers are instantaneous values of the analog signal at each sampling interval

• After being sampled, narrow pulses are generated

• These sample values are either amplitude, width of position of the pulse

DISCRETE BASEBAND

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• Analog modulation (AM) and frequency modulation (FM) are example of continuous wave (CW) modulation

• In pulse modulation, analog signals are sampled

• The samples are varied according to a parameter of the pulse waveforms, e.g. amplitude, width, or position of the pulse

• Sampling theorem says that a low pass modulating signal that is band-limited to fm Hz has to be sampled at a rate of at least 2fm samples per second

PULSE MODULATION


• Natural sampling of a band-limited signal:

A REVIEW OF SAMPLING

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A REVIEW OF SAMPLING

• Sampling of f(t) results in the general of spectral replicas at multiples of the sampling frequency

• Sampling pulses do not have to be rectangular in shape and the particular choice of the pulse form changes the shape of the envelope of the spectrum Fs(w)

• The original signal can be recovered from the sampled signal fs(t) by using an ideal low pass filter

• In natural sampling, the slopes of the pulse tops have the same shape as the slopes of the modulating signal

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• In PAM, the amplitude of a train of constant-width pulses is varied in proportion to the sample values of the modulating signal

• In PAM the pulse tops are flat

• The PAM signal can be obtained by first multiplying the signal by a train of impulses, and then applying the result to an LTI filter with unit impulse response q(t)

PULSE AMPLITUDE MODULATION (PAM)

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• Spectrum of a PAM signal:


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• Spectral analysis

• The output of the LTI filter in the time domain is the convolution of fs(t) and the filter impulse response q(t)

• In frequency domain this is equivalent to multiplication:


1 2( )s

n

F F nT T

w w

( ) ( )T

n

p t t nT

( ) ( ) ( ) ( ) ( )s

n n

f t f t t nT f nT t nT

1 2( ) ( ) ( )s

n

F Q F n QT T

w w w w




• Note that this spectral density is not the same as the repeated spectral density of the original signal only with a gain variation as we had for natural sampling

• In PAM, a point to point multiplication of Fs(w) and Q(w) is obtained, hence the spectral density of the output of the filter is not the same as the original signal

• The spectral distortion introduced is a function of the pulse shape

• At low frequencies and with narrow pulse width, it is not severe

• Why flat top PAM? Because the shape of the pulses to is not needed to convey information, and rectangular pulses are easy to generate

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Advantage of PAM over CW Modulation

• With long distance transmission, repeaters are necessary

• In analog CW modulation systems, the repeaters must amplify the whole signal faithfully, thus additive noise effects are compounded

• In PAM, the pulse shape is not important, hence the repeater can re-generate the pulses rather than amplifying them (e.g. make the area of the regenerated pulse proportional to the area of the received pulse over a fixed interval of time)

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Signal Recovery from PAM Waveform

• One possibility is to sample the PAM waveform with a periodic train of very narrow pulses (synchronised with the PAM signal) and smooth the outcome with a low-pass filter

• Note that f(t) cannot be recovered using a LPF, because the low frequency PAM spectrum is Fs(w)Q(w)

• We need a filter with the transfer function Q-1(w), and since f(t) is bandlimited, it is sufficient to synthesize the inverse filter only over a very limited frequency range

• The LPF and the inverse filter can be combined:

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•The filter function:

•This technique of correcting the frequency response of a system for a known distortion is called equalisation (used over telephone lines as well)

•For a rectangular pulse shape, a sinc function is used for Q(w):

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1( ) | |( )

0 elsewhere

mQH

w w ww

T = sampling

period




Smaller values of the pulse width, t, moves the zero crossing further away from the origin and Q-1(w) tends to become flatter within the limited bandwidth of the LPF

T and t are independents and t/T is a measure of the flatness of Q(w) and Q-1(w) within the bandwidth of LPF

If t/T <= 0.1, the maximum difference between the ideal response and the actual response over the required range is less than 1%

Hence, if t /T <= 0.1, equalisation for PAM can usually be neglected

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• PAM uses narrow pulse widths, spare space is available and can be allocated to pulses from other signals in a definite time sequence (example for 2 channels given below)

TIME-DIVISION MULTIPLEXING (TDM)



• Principles of TDM applies to other types of pulse modulation as well

• The commutator determines the synchronisation and the sequence of the channels

• A pulse generator produces the narrow pulses required to drive the sampler

• The clock determines the timing of the overall system • As an example, assume that both input signals are lowpass

and bandlimited to 3 kHz • Sampling theorem gives a minimum sampling rate of 6 kHz

for each signal (2 x highest frequency component of the signal)

• There are 2 signals, therefore a minimum clock rate of 12 kHz for the system



• 2-channel TDM PAM:

• The bandwidth of this signal is high (theoretically infinite),

but how much transmission bandwidth is really needed for correct demodulation?

• Assume that only amplitude information is required • For an ideal LPF which bandlimits the signal to Bx Hz,


1

2x

x

BT




• Note that this bandwidth only guarantees that independent amplitude information is present in the signal, but pulse shapes may not closely resemble the pulses generated by the sampler, pulse shape recognition will require additional bandwidth

• Assume that the ideal LPF has the transfer function:

Note: W=2Bx

• The impulse response of this filter is

/ | |<W( )

0 elsewherex

WH

ww

1( ) ( ) sinc( )x xh t F H Wtw



The arrival of each impulse of the TDM signal at the input of the filter produces the impulse response of the filter at the output

The effective width of this impulse response is 2/W=1/Bx



Hence the filter output is the superposition of all sinc functions:

Because the spacing between successive samples is 1/2Bx , contributions from all adjacent samples are exactly zero at the correct sampling instant

When all input signals have the same bandwidth, the efficiency of PAM is almost the same as SSB AM

With different bandwidths, the efficiency reduces 23



Typical TDM PAM receivers, with or without sample-and-hold circuit

Synchronisation of the clock and the commutator is necessary between the transmitter and the receiver

Periodically send a pulse that exceeds the height of any other pulses in the transmission, use this at Rx for synchronisation

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INTERSYMBOL INTERFERENCE

So far interference between adjacent channels has been ignored due to the following assumptions:

each pulse passing through the ideal LPF at the receiver generates a perfect sinc function

at correct sampling instances, only the sinc function from the current pulse is non-zero, all other sinc functions are zero at that point

perfect synchronisation between the transmitter and the receiver (ie. knowing the perfect sampling instance at the receiver)

the transmission medium behaves like an ideal LPF

In reality, none of the above assumptions are true, and we get some cross talk between adjacent channels, called intersymbol interferece (ISI)



The available options are:

increase the transmission bandwidth (wasteful)

design the transmission filters and the receiver such that the waveform received have minimum ISI within as small transmission bandwidth as possible (way to go!)

Analysis of the received PAM signal y(t):

For a given time t = kT,

( ) ( ) ( ) : an integer

( ) : pulse waveform : sampling period

: sampled values of the input ( ) : additive noise

m

m

m

y t a x t mT n t m

x t T

a n t

( ) (0) ( ) ( )k m

m k

y kT a x a x k m T n t




1st term: sampled value of the kth pulse (symbol) that has been transmitted

2nd term: interference from adjacent symbols at t = kT, can be eliminated by designing the received pulse shape x(t) to satisfy the following condition:

Rather than the unrealizable sharp LPF, a filter that has gradual roll-off, and yet still has an impulse response that has uniformly spaced zeros, distance T apart (except the centre) is needed

The raised cosine characteristics satisfies this condition (it belongs to a class of waveforms known as Nyquist waveform)

1

( )0

m kx k m T

m k



The raised-cosine function in frequency domain:

The corresponding impulse response is:

0 | | (1 )

( ) 1 sin | | (1 ) | | (1 )2 2

0 (1 ) | |

excess bandwidth used divided by min. Nyquist bandwidth

/

T W

TX W W W

W

W

W T

w

w w w

w

2

cos( )( ) sinc( )

1 (2 / )

Wtx t Wt

Wt

impulse response of the

ideal LPF, and retains the original zero crossings

more gradual roll-off in the frequency characteristics



Plot of raised-cosine pulse:

With a =1, x(t) has zeros halfway between sampling times and the oscillatory tails of x(t) are smallest, hence lower ISI with timing errors in sampling, but it requires twice the bandwidth

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spectrum

impulse response


a is known as the roll-off factor. For full-cosine roll-off characteristics (a =1),

Larger values of a (>1) lead to faster decaying pulses so that receiver synchronisation will be less critical, but this will be at the expense of added bandwidth:

The filters we have designed so far assume impulse inputs, but PAM signals have finite-width (narrow) pulses

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1 cos | | 2( ) 2 2

0 elsewhere

TW

X W

ww

w

1

2x

x

BT



Designating Q(w) to the spectrum of these finite-width pulses, the magnitude of the frequency transfer function of the required shaping filter is:

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( )( )

( )

XH

Q

ww

w


Example

• Twenty PAM signals each band-limited to 3 kHz and sampled at 8 kHz, are time multiplexed prior to transmission. Choose a raised-cosine filter characteristic that will permit transmission of this multiplexed signal within an absolute maximum bandwidth of 120 kHz.

• Solution:

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1

2x

x

BT

616.25 10

20 8xT

kHz

6 3

2 1

2 6.25 10 120 10 1

1.5 1

0.5

x xT B



• An alternative to PAM is to vary some parameter in the timing of each pulse to convey the information, ie. pulse timing modulation

• One option is to change the width of fixed amplitude pulses proportional to the values of f(t) at the sampling instants, hence called pulse width modulation (PWM)

• Another option is to change the position of fixed amplitude, fixed width pulses proportional to the values of f(t) at the sampling instants, hence called pulse position modulation (PPM)

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OTHER ANALOG PULSE MODULATIONS


• Plots of analog pulse modulations:

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Amplitude of f(t) at sampling intervals

Width of generated pulse is proportional to the amplitude at sampling intervals

The pulse in generated from a reference point proportional to the amplitude at sampling intervals



• In PWM : a minimum width t0 is assigned to the minimum of f(t) variation from t0 is proportional to f(t) (modulation

constant K1) pulse duration must be shorter than the time slot

allocated to a particular sample (an additional guard time tg is allowed)

the average value of a PWM waveform varies directly with the modulation

• In PPM : a minimum pulse delay is assigned to the minimum of

f(t) variation from the minimum pulse delay is proportional

to f(t) (modulation constant K1)

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ELEC3540 Lecture 2 Analog Baseband Transmission

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