ece 4710: lecture #6 1 bandlimited signals bandlimited waveforms have non-zero spectral components...

ECE 4710: Lecture #6 1

Bandlimited Signals

Bandlimited waveforms have non-zero spectral components only within a finite frequency range

Waveform is absolutely band limited to B Hertz if

Waveform is absolutely time limited if

Theorem: Absolutely time limited signals have infinite BW (e.g. ) Absolutely band limited signals have infinite time

representation (e.g. )

BftwfW ||for 0)]([)(

Tttw ||for ,0)(

Tt

)2cos( tf


Real Signals

Physically realizeable signals must be time limited and therefore must have infinite bandwidth?? Consequence of mathematical model

Real signals may not be theoretically bandlimited but they are practically bandlimited Amplitude spectrum is negligible (no significant power)

beyond a certain frequency range, e.g. signal power level falls below noise power level

Example: 99.9% of all the power is contained within a frequency range of ± 2 MHz practical signal BW


Sampling Theorem

Any physically bandlimited waveform can be reconstructed

without error if the sampling frequency is Nyquist criterion

For many waveforms a perfect reconstruction is not practical due to data rate and channel bandwidth limitations

For an approximate signal reconstruction over a restricted period of time, T0 , the minimum number of samples needed is

Pre-filtering signal waveform to reduce occupied signal bandwidth and the number of required samples

Bf s 2

000

min 2/1

BTTff

TN s

s


Impulse Sampling & DSP

For digital systems we must represent a time domain signal using discrete # of samples

Impulse sampled waveform w(t) ws(t)

How does sampling affect the frequency spectrum of a signal? Answer depends on the sampling rate fs

ss

nss

nss

fT

nTtnTwnTttwtw

/1 e wher

)()()()()(


Impulse Sampling & DSP

What is Fourier Transform of ws(t) ? FT of impulse train (Table 2-2)

Multiplication in time is convolution in frequency

Spectrum of impulse sampled signal is the spectrum of the unsampled signal that is repeated every fs Hz Fundamental principle of Digital Signal Processing (DSP)

tTs

f

fs 2fs-fs-2fs 0

. .

.. . .

......

n

ssn

ss

ss nffWT

nffT

fWtwfW )(1

)(1

)()]([)(


Nyquist Sampling


Nyquist Sampling

Since impulse sampled spectrum has multiple copies of baseband signal spectrum spaced by fs the possibility exists that the spectral copies could overlap and interfere with each other If fs 2 B then the replicated spectral copies do not

overlap sampling at the “Nyquist rate”

If fs < 2 B then the time waveform is “undersampled” causing spectral overlap» Overlap is called “aliasing” or “spectral folding”» Significant distortion of original waveform


What to do if signal BW is too large causing fs < 2B ?

Pre-filter signal to reduce occupied signal BW Distortion still occurs but result is better than aliasing

Must be done with BW signals, e.g.

Under-Sampling

Tt


Signal BW

Spectral bandwidth (BW) of signals important for two main reasons:1) Available frequency spectrum is very congested

» Wireless services & applications increasing dramatically» Spectrally efficient communication systems needed to conserve

available spectrum and maximize # of users

2) Communication system must be designed with enough

bandwidth to capture desired signal & reject unwanted

signals and noise How do we define a signal’s BW?

Many different ways and all are useful


Signal BW

Given the variety of methods for measuring signal BW, care must be taken : To ensure consistent application for S/N calculations

when comparing different signals and systems Engineering definitions of signal BW deal with

positive frequencies only ( f > 0) Real waveforms and filters have magnitude spectrum that

are symmetric about origin, e.g. they contain + and – f Signal BW must be f2 – f1 where f2 > f1 > 0

» For baseband signals f1 = 0

» For bandpass signals f1 > 0 and f1 < fc < f2

fc is the carrier frequency of modulated waveform


BW Definitions

Six Engineering Definitions Absolute BW = f2 – f1 where the spectrum is zero outside

the interval f1 < fc < f2

3-dB or Half-Power BW = f2 – f1 where f1 & f2 correspond to frequencies where magnitude spectra are

Equivalent Noise BW = width of fictitious spectrum such that the power in that rectangular band is equal to the power associated with the actual spectrum over positive frequencies

5.0|)(||)(|

so and 2/1|)(||)(|2

22

1

21

fWfW

fWfW


Six Engineering Definitions Null-to-Null BW = f2 – f1 where f2 and f1 are the first nulls

in the envelope of the magnitude spectrum above and below f0 (for bandpass signals)

» f0 is the frequency where the magnitude spectrum is maximum

» For baseband signals f1 = 0 First Null BW (FNBW)

BW Definitions

f

PSD

FNBWf

PSD

N-to-N BW

f0


BW Definitions

Six Engineering Definitions

Bounded Spectrum BW = f2 – f1 where outside the band f1 < f < f2 , the PSD must be down by at least a certain amount, e.g. 50 dB below the max PSD value

Power BW = f2 – f1 where f1 < f < f2 defines the frequency band in which 99% of the total power resides» Similar to FCC definition of occupied BW where power above the

upper band edge f2 is 0.5% and the power below the lower band edge f1 is 0.5%


BW Definitions

Legal BW Definition in U.S. defined by FCC FCC Bandwidth

» Sec. 21.106 of the FCC Rules and Regulations : “For operating frequencies below 15 GHz, in any 4 kHz band, the center frequency of which is removed from the assigned frequency by more than 50 percent up to and including 250 percent of the authorized bandwidth, as specified by the following equation, but in no event less than 50 dB”:

» Attenuation > 80 dB is NOT requiredMHzin BW authorized

frequencycarrier thefrom removedpercent

leveloutput mean thebelow dB)(in n attenuatio

where)(log10)50(8.035 10

B

P

A

BPA


Legal definition & equation define a spectral mask:

Signal spectrum must be the values given by the formula at all frequencies

FCC Bandwidth


BPSK Signal BW

Binary Phase Shift Keying (BPSK) signal

m(t) is serial binary (1) modulating waveform For real data m(t) is random but we will assume a mathematical

“worst-case” (wide BW) model where 1 transitions occur the most frequent:

Data Rate = R = 1 / Tb (bps = bits per second)

)2cos()()( tftmts c


BPSK Signal BW

Spectral shape is Sa2 function at fc

Discrete line spectrum for deterministic worst-case model Continuous spectrum for random data (all frequencies present)

22

)()(sin

41

)()(sin

41

)(

cb

cbb

cb

cbbBPSK ffT

ffTT

ffTffT

Tf

P


BPSK Signal BW

BW Definition Bandwidth BW for R = 9.6 kbps

Absolute 3-dB 0.88 R 8.5 kHz

Eq. Noise 1.0 R 9.6 kHz

Null-to-Null 2.0 R 19.2 kHz

Bounded (50 dB) 201.0 R 1.93 MHz

Power (99%) 20.6 R 197 kHz

**3-dB BW is smallest of all measures

ece 4710: lecture #6 1 bandlimited signals bandlimited waveforms have non-zero spectral components...

Documents