energy spectral density measurement of energy as a f() of freq PSIs(f) = |S(f)|^2 autocorrelation is matching of a sig with delayed version of itself, for a real valued energy sig s(t) PSIs(tau) = int(s(t)s(t-tau)...) ex3-5: a waveform s(t) has mag spectrum: S = rect(f/400) + 0.5rect(f/200) + 0.5rect(f/100) find energy spectral density PSIs(f) = mag^2(S) = rect(f/400) + 1.25rect(f/200) + 1.75rect(f/100) find the energy in the signal: Es = int(PSIs(f)df) = 2int<0,inf>(PSIs(f)df) = 2(4*50 + 2.25*50 + 1*100) = 825VV/s s(t) is passed through an ideal low pass filter w/ co freq of 75hz, find energy in output signal: E = 4*100 + 2.25*50 = 400+22.5*5 = 512.5 percent energy passed by filter: 512.5/825 = 62.blah % noise electrical noise is any undesirable electrical energy that falls within th epassband of the signal 2 general categories of noise: uncorrelated: present with or without signal atmospheric, extraterrestrial, man-made, interference, shot noise (caused by random arribal of carriers at the output element of an electronic device) thermal (proportional to the product of BW and temp, N = KTB, N is noise power in watts, K is boltzmann's constant, T in kelvin, white noise/const pwr density) correlated: noise factor: F = input SNR/ouput SNR noise figure: NF = 10log(F) signal BW: width of freq band that contains a sufficient amount of the signal's freq components to reproduce the sig without an unacceptable amount of distortion Wn = Ps/Gs(fc), Ps is total sig power over all freqs and Gs is value of Gs(f) at band center null to null BW = width of main spectral lobe fractional power continament BW: occupied BW is the band that leaves exactly 0.5% of the sig power above the upper band limit and exactly 0.5% of the sig power below the lower band limit, thus 99% of the sig power is inside the occupied band bounded pwr spectral density: everywhere outside the specified band, Gs must have fallen at least to a certain level below that found at the band center, typical attenuation levels are 35 or 50dB ex3-6: fractional power containment BW s(t) = e^(-at)u(t) find sig energy Es = int(s^2(t)dt) = int<0,inf>(e^(-2at)dt) = 1/2a find BW that contians 95% of the signal energy first, need fourier transform: S(f) = 1/(a+j2fpi) PSIs = abs^2(S) = 1/(aa + 4pi*pi*ff) at what freq W does the signal PSIs contain 95% (0.95/2a) of the energy? 2int<0,w>(df/(aa + 4ffpi*pi)) = 2int<0,w>((1/(4pi*pi))*df/(aa/(4pi*pi) + ff)) = 1/api * arctan(2Wpi/a) == .95/2a -> 2Wpi/a = tan(.95pi/2) = 12.706 -> W = 2.022a baseband vs carrier communications term baseband is used to designate the freq band of the original message signal in baseband comms, message sigs are directly transmitted without any mods baseband sigs generally have significant low freq components so multiple sigs will interfer if transmitted over a common channel in carrier comms, modulationi is used to shift the freq spectrum of a sig in analog modulation, one of the basic parameters (amp, freq, phase) of a sinusoidal carrier of freq fc is varied linearly with baseband sig m(t) double sideband amp modulation AM is characterized by an infomation bearing carrier amp A(t) that is a linear function of the baseband message m(t) if m has FT M, m(t)cos(2tfc*pi) <-> ex4-1: m(t) -----> (X) -> m(t)cos(2tfc*pi) == x(t) cos(2tfc*pi)-^ assume m has finite BW multiply the two sigs gives spectrum of m shifted to fc and -fc x(t) -----> (X) -> m(t)cos^2(2tfc*pi) = m(t)/2(1 + cos(2tpi*fc)), use filter to remove the high freq term cos(2tfc*pi)-^