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##### Lecture 2

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**Lecture 2**• Review of Fourier Series • Review of Fourier Transforms • Example • Spectrum and bandwidth • Relationship between data rate and bandwidth • Analog and digital data transmission ECS152A – Computer Networks**Spectrum & Bandwidth**• Spectrum • range of frequencies contained in signal • Examples: (1) for s(t) = sin(2f1t) + 1/3 sin(2p(3f1)t) the spectrum is from f1 to 3f1 (2) for s(t) = square wave of time period T (=1/f) the spectrum is f to • Absolute bandwidth • width of spectrum • Examples: (1) absolute bandwidth is 2f1 (2) absolute bandwidth is ECS152A – Computer Networks**Spectrum & Bandwidth**• Effective bandwidth • Often just bandwidth • band of frequencies containing most of the energy • DC Component • Component of zero frequency ECS152A – Computer Networks**Power Spectrum**• The function s(t) specifies the signal in terms of voltage or current • The average power of a time limited signal is given by • P= (1/t2 - t1) t1->t2 |x(t)|2 dt • For periodic signal the average power in one period (T) is given by • P= (1/T) 0->T |x(t)|2 dt ECS152A – Computer Networks**Power Spectrum**• We would like to know the signal power as a function of the frequency -- power spectral density • For periodic signals the power spectrum is discrete • Pj = power through the first j harmonics = (1/4) C02+ (1/2) n=1 ->j |cn|2 • For aperiodic signal the power spectrum is continuous • P = power contained in a band of frequencies f1 and f2 =2 f1 -> f2 S(f) df ECS152A – Computer Networks**Power Spectrum**• Half-power bandwidth • Range of frequencies at which S(f) has dropped to half of its maximum value of power • Paseval’s Power Theorem • The normalized value of the mean power of a periodic signal is equal to the sum of squared amplitudes of all harmonic components of signal s(t). ECS152A – Computer Networks**Relationship Between Data Rate and Bandwidth**• Just like the signal has a bandwidth, any transmission system has a bandwidth • the transmission medium accommodates a limited band of frequencies • This limits the data rate that can be carried • Consider a periodic square wave s(t) of period T • Using Fourier series the square wave can be expressed as • s(t) = (4/k) n odd & n=1 -> (1/n)sin(2nf1t) • where f1=1/T is the period of the signal ECS152A – Computer Networks**Relationship Between Data Rate and Bandwidth**• This waveform has infinite number of harmonics and hence infinite bandwidth • The peak amplitude of the kth component is 1/k that of the fundamental frequency • Most of the power will be concentrated in the first few harmonics ECS152A – Computer Networks**Case 1**• Let us consider that the square wave is very well approximated by the first three harmonics. I.e., • s(t) = 4/p (sin 2pft + 1/3 sin 2p 3f t + 1/5 sin 2p 5f t) ECS152A – Computer Networks**High**SNR t t t noise signal + noise signal Low SNR t t t Average Signal Power SNR = Average Noise Power SNR (dB) = 10 log10 SNR signal + noise signal noise ECS152A – Computer Networks Figure 3.12**High**SNR t t t noise signal + noise signal Low SNR t t t Average Signal Power SNR = Average Noise Power SNR (dB) = 10 log10 SNR Signal to Noise Ratio signal + noise signal noise ECS152A – Computer Networks Figure 3.12**Analog and Digital Data Transmission**• Data • Entities that convey meaning • Signals • Electric or electromagnetic representations of data • Transmission • Communication of data by propagation and processing of signals ECS152A – Computer Networks**Data**• Analog • Continuous values within some interval • e.g. sound, video • Digital • Discrete values • e.g. text, integers ECS152A – Computer Networks**Analog (Stream) Date**• Voice, video occur in steady stream • The data can take continuous values ECS152A – Computer Networks**W**W W W Red Component Image Green Component Image Blue Component Image Color Image = + + H H H H Total bits before compression = 3xHxW pixels x B bits/pixel = 3HWB Block (Digital) Data • Examples • Text files • Scanned color documents ECS152A – Computer Networks Figure 3.1**176**(a) QCIF Videoconferencing 144 @ 30 frames/sec = 760,000 pixels/sec 720 (b)Broadcast TV 480 1920 (c) HDTV 1080 Video Data @ 30 frames/sec = 10.4 x 106 pixels/sec @ 30 frames/sec = 67 x 106 pixels/sec ECS152A – Computer Networks**Converting Analog Data to Digital Data**• Sampling • What should be the sampling rate? • Nyquist rate: Twice the maximum bandwidth of the signal • Digitize the sample • Quantization error ECS152A – Computer Networks**Transmission Medium**• Type • Guided medium - e.g. twisted pair, optical fiber • Unguided medium - e.g. air, water, vacuum • Characteristic • Simplex • signal is transmitted in only one direction • Half-duplex • both station may transmit but only one at a time • Full-duplex • both stations may transmit simultaneously. The medium is carrying signals in both directions. ECS152A – Computer Networks**(a) Analog transmission: all details must be reproduced**accurately Received Sent Received Sent • e.g digital telephone, CD Audio Analog and Digital Transmission • e.g. AM, FM, TV transmission (b) Digital transmission: only discrete levels need to be reproduced ECS152A – Computer Networks**Transmission segment**Destination Source Repeater Repeater A Generic Long-Distance Link • Over distance • Signal is attenuated • Noise (thermal and electromagnetic) noise gets added. • Repeaters regenerate signal ECS152A – Computer Networks Figure 3.7**Recovered signal**+ residual noise Attenuated & distorted signal + noise Amp. Equalizer Repeater An Analog Repeater • Amplify the signal • Eliminate distortion using the equilizer • Different frequency components are • Attenuated differently • Delayed differently • Noise accumulates ECS152A – Computer Networks**Decision Circuit.**& Signal Regenerator Amplifier Equalizer Timing Recovery A Digital Repeater • Amplifier and equalizer mitigate some of the channel distortion • Goal is not to regenerate the signal but to determine the original pulse • Regenerate a fresh pulse • Noise does not accumulate ECS152A – Computer Networks**Advantages of Digital Transmission**• Digital technology • Low cost LSI/VLSI technology • Data integrity • Longer distances over lower quality lines • Capacity utilization • High bandwidth links economical • High degree of multiplexing easier with digital techniques • Security & Privacy • Encryption • Integration • Can treat analog and digital data similarly ECS152A – Computer Networks