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
ECE 4710: Lecture #6 2
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
ECE 4710: Lecture #6 3
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
ECE 4710: Lecture #6 4
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
)()()()()(
ECE 4710: Lecture #6 5
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
)()]([)(
ECE 4710: Lecture #6 6
Nyquist Sampling
ECE 4710: Lecture #6 7
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
ECE 4710: Lecture #6 8
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
ECE 4710: Lecture #6 9
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
ECE 4710: Lecture #6 10
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
ECE 4710: Lecture #6 11
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
ECE 4710: Lecture #6 12
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
ECE 4710: Lecture #6 13
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%
ECE 4710: Lecture #6 14
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
ECE 4710: Lecture #6 15
Legal definition & equation define a spectral mask:
Signal spectrum must be the values given by the formula at all frequencies
FCC Bandwidth
ECE 4710: Lecture #6 16
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
ECE 4710: Lecture #6 17
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
ECE 4710: Lecture #6 18
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