Report

Noise on Analog Systems ECE460 Spring, 2012 AM Receiver H(f) r t 1 Received Signal s t nW t H(f) -f2 -f1 f1 f2 f cos 2 f c t y t 1 -W W f • s(t) is the transmitted signal • Let M(t) be a random process representing the information bearing signal. m(t) will denote a sample function of M(t). M(t) is assumed zero mean WSS with autocorrelation function RM(τ) and power spectral density SM(f). M(t) is assumed a low-pass signal or a baseband signal with spectral content limited to W Hz, i.e., S M f 0 for f W . and the signal power is W PM S M f df W • nW(t) is a sample of zero mean, white noise with power spectral density N0/2. • The received signal after the ideal BPF is r t s t n t where n(t) is narrow-band noise. 2 Noise in the Receiver For DSB-SC Amplitude Modulation, f1 f c W f2 fc W And the PSD of n(t) is SN(f) N0 2 -fc -W -fc fc -W -fc+W fc+W fc From our work on bandpass processes, n(t) can be broken into in-phase and quadrature components n t nc t cos 2 f ct ns t sin 2 f ct where nc(t) and ns(t) are uncorrelated processes, i.e. E nc t ns t 0 t and Furthermore, and S nc f S ns f Rnc Rns given by N0 -W W f 3 Evaluation of DSB-SC Recall that the transmitted signal is s t Ac m t cos 2 f ct c The received signal, r(t), after the ideal BPF filter is r t s t n t Ac m t cos 2 f c t c nc t cos 2 f c t ns t sin 2 f ct where the ideal bandpass filter H(f) at the receiver’s input is 1, Hf 0, f fc W Otherwise and W is the bandwidth of the information process m(t). Find the demodulated r(t): 4 DSB-SC SNR at Output Assuming synchronous demodulation (e.g., c ), find y(t). Power of the received signal at the receiver’s output: Power of the noise at the receiver’s output: SNROut desired signal power at receiver output noise power at receiver output 5 DSB-SC SNR at Input To transmit out signal m(t), we used a transmitter with power equal to the power of s(t) given by Sin This was also considered the power of the signal at the receiver’s input. The noise power at the receiver input calculated for the message bandwidth is N in SNRin transmitted signal power receiver noise at signal bandwidth 6 Conventional AM (DSB) Recall the transmitted signal is s t Ac 1 a mn t cos 2 f ct where |m(t)| ≤ 1. If we follow the methodology in DSB-SC assuming synchronous modulation and θ = 0 without any loss of generality, then the output of the low-pass filter is yl t 12 Ac 1 a mn t 12 nc t The dc component, Ac/2, is not part of the message and must be removed. The output after a dc blocking device is: y t 12 Ac a mn t 12 nc t Find the SNR at the receiver’s output and input. SNRout 7 SSB Recall the transmitted signal is s t Ac m t cos 2 f ct Ac m t sin 2 f ct The received signal is r t Ac m t nc t cos 2 f ct Ac m t ns t sin 2 f ct Again assuming synchronous demodulation with perfect phase, the output after the LPF is y t 12 Ac m t 12 nc t Find: Sout N out SNRout SNRin 8 Example The message process M(t) is a stationary process with the autocorrelation function RM 16sinc 2 10, 000 It is also known that all the realizations of the message process satisfy the condition max |m(t)|=6. It is desirable to transmit this message to a destination via a channel with 80-dB attenuation and additive white noise with power-spectral density Sn(f) = N0/2 = 10-12 W/Hz, and achieve a SNR at the modulator output of at least 50 dB. What is the required transmitter power and channel bandwidth if the following modulations schemes are employed? 1. DSB-SC 2. SSB 3. Conventional AM with modulation index equal to 0.8. 9 Angle Modulation Effect of additive noise on modulated FM signal • • Amplitude Modulation vs Angular Modulation Importance of zero-crossing -> instantaneous frequency Approximate • Block diagram of the receiver Receiver s t nw t Bandpass Filter r t s t n t Angle Demodulator y (t ) S N 0 Lowpass Filter nw t zero mean Gaussian white noise s t A c cos 2 f c t t transmitted signal k pm t t t 2 k f m d PM FM Bandpass filters limits noise to bandwidth of modulated signal • n(t) is bandpass noise n t n c t cos 2 f ct ns t sin 2 f ct Or, in Phasor form n t Vn t cos 2 f ct n t where Vn t n c2 t ns2 t n t n t arctan s n t c 11 Phasor Analysis Assumption: Vn t Ac r t 12 Solve for SNR Found demodulated signal y(t) • Composed of signal and additive noise k p m t Yn t PM y t 1 d k m t Yn t FM f 2 dt Yn t 1 ns t cos t nc t sin t Ac RYn t , t 1 Rn E cos t t Ac2 c • Assumption: m(t) is a sample function of a zero mean stationary Gaussian process with autocorrelation function RM(τ). • What about t ? k pm t Recall: t t 2 k f m d PM FM 13 Typical Plots 14 Noise Power Spectrum at Demodulated Output S nout f PM FM S nout N0 PM A2 , c f N 0 f 2 , FM Ac2 15 Noise and Signal Power at Output (LPF) Noise Power at LPF Pnout S f df nout 2W N 0 A2 , PM c 3 2W N 0 , FM 3 Ac2 To Determine Power Out of Signal, recall k p m t Yn t PM y t 1 d k m t Yn t FM f 2 dt Signal Power at LPF PSout k p2 PM , PM 2 k f PM , FM 16 SNR for Angle Modulation Therefore, SNRout SNRout PSout Pnout k p2 Ac2 2 2 2 3k f Ac 2W 2 PM N 0W PM PM , FM N 0W Using Modulation Indexes p k p max m t t f kf W max m t t And denoting SNRb power of transmitted signal power of noise at message bandwidth Ac2 / 2 N 0W Then SNRout SNRb g 17 SNRout / SNRb 1. Proportional to modulation index squared 2. Increasing improves SNR gain at the expense of bandwidth expansion 3. The maximum possible SNR gain improvement is exponential as can be shown using Shannon theory 4. We cannot increase without limit sent at some point our results will not be valid since they are only approximate results 5. FM, like any other nonlinear modulation technique, exhibits a threshold effect and performance. Above certain SNRb , the SNRout is proportional to 2 SNRb . Below the threshold, SNRout maybe worse than SNRb. 18 SNRout / SNRb 6. In AM, increasing Ac increases SNRout since the received message is proportional to Ac. Here, increasing Ac also increases SNRout but through a different mechanism. Here increasing Ac reduces the amount of noise that affects the message signal. 7. To compensate for the high noise PSD at high frequencies and FM, the PSD of the signal is pre-emphasized in the transmitter to increase its immunity to noise at high frequencies and it is then the emphasized at the receivers output. 8. It can be shown that at threshold, we have for FM systems SNRb 20 1 Given the received power of the modulated signal, this relation gives us the max which ensures that the system works above threshold. Another restricting factor results from Carlson’s rule Bc 2 1 W Then, give and receive power and channel bandwidth SNRb Bc min , 1 20 2W 19 Example Consider an FM broadcast system with parameter k f 75 103 , and W 15 kHz. Assuming PM 12 , find the output SNR and calculate the improvement (in dB) over the baseband system. 20 Pre-emphasis & De-emphasis Filters 21