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. )

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, T 0, the minimum number of samples needed is  Pre-filtering signal waveform to reduce occupied signal bandwidth and the number of required samples

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)  w s (t)  How does sampling affect the frequency spectrum of a signal?  Answer depends on the sampling rate f s

ECE 4710: Lecture #6 5 Impulse Sampling & DSP  What is Fourier Transform of w s (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 f s Hz  Fundamental principle of Digital Signal Processing (DSP) t TsTs f  fsfs 2f s -f s -2f s 0...

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 f s  the possibility exists that the spectral copies could overlap and interfere with each other  If f s  2 B then the replicated spectral copies do not overlap  sampling at the “Nyquist rate”  If f s < 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 f s < 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

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 f 2 – f 1 where f 2 > f 1 > 0 »For baseband signals  f 1 = 0 »For bandpass signals  f 1 > 0 and f 1 < f c < f 2  f c is the carrier frequency of modulated waveform

ECE 4710: Lecture #6 11 BW Definitions  Six Engineering Definitions  Absolute BW = f 2 – f 1 where the spectrum is zero outside the interval f 1 < f c < f 2  3-dB or Half-Power BW = f 2 – f 1 where f 1 & f 2 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

ECE 4710: Lecture #6 12  Six Engineering Definitions  Null-to-Null BW = f 2 – f 1 where f 2 and f 1 are the first nulls in the envelope of the magnitude spectrum above and below f 0 (for bandpass signals) » f 0 is the frequency where the magnitude spectrum is maximum »For baseband signals f 1 = 0  First Null BW (FNBW) BW Definitions f PSD FNBW f PSD N-to-N BW f0f0

ECE 4710: Lecture #6 13 BW Definitions  Six Engineering Definitions  Bounded Spectrum BW = f 2 – f 1 where outside the band f 1 < f < f 2, the PSD must be down by at least a certain amount, e.g. 50 dB below the max PSD value  Power BW = f 2 – f 1 where f 1 < f < f 2 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 f 2 is 0.5% and the power below the lower band edge f 1 is 0.5%

ECE 4710: Lecture #6 14 BW Definitions  Legal BW Definition in U.S. defined by FCC  FCC Bandwidth »Sec 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 required

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 / T b (bps = bits per second)

ECE 4710: Lecture #6 17 BPSK Signal BW  Spectral shape is Sa 2 function at  f c  Discrete line spectrum for deterministic worst-case model  Continuous spectrum for random data (all frequencies present)

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) R 1.93 MHz Power (99%) 20.6 R 197 kHz **3-dB BW is smallest of all measures