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Transcript
Signals & Waves A. Eskandarian 7/11/2017 1 Signals & Waves We have learned how to construct wave packets for 3-dimensional waves similar to 1-dimensional waves. We’ll revisit the frequency spectrum We’ll learn about analog Vs digital signals We will use the Sampling theorem for digitization of analog signals 7/11/2017 2 Signals & Waves Fourier series allows us to construct (represent) any periodic function in terms of sums of Sine and Cosine functions (possibly infinite series) with differing amplitudes and arguments that are multiples of a fundamental frequency. It also allows us to represent any general signal being composed of frequencies with continuous values in a given bandwidth. 7/11/2017 3 Signals & Waves In the first case the frequency spectrum is discrete. (i.e., the graph of Amplitude VS Frequency) In the second case the curve is a continuous curve. 7/11/2017 4 Signals & Waves Amplitude f0 3f0 5f0 7f0 frequency Frequency spectrum for the discrete case (maybe infinite number of component frequencies) 7/11/2017 5 Signals & Waves Bandwidth = (f2- f1) Amplitude f1 f2 frequency Frequency spectrum for the continuous case (infinitely many frequencies between f1 and f2, f1< f < f2) 7/11/2017 6 Signals & Waves The following general observations may be made in conjunction with the frequency spectrum of a given function: 1. A discrete frequency spectrum belongs to a signal that is periodic and can be represented by a Fourier series (either finite or infinite) 2. A continuous frequency spectrum mostly limited to a finite bandwidth is that of a general localized signal that does not necessarily appear to exhibit periodic behavior. 7/11/2017 7 Signals & Waves If a general analog signal of the second type is given, and sent by a transmitter. It must be reproduced exactly at the receiver site to convey the information it carries. Distortion, attenuation, and noise will transform its profile. Is it possible to reproduce this signal without having to know the amplitude at every given time? YES 7/11/2017 8 Signals & Waves Sampling Theorem (Shannon) tells us that if the frequency spectrum of the signal is limited to Bandwidth = (f2- f1) then a sampling rate of 2(f2- f1) will faithfully reproduce the original signal. In other words, all the information present in the original signal can be reconstructed by having knowledge of only those sample points. 7/11/2017 9 (a) 7D/2 5D/2 3D/2 D/2 -D/2 -3D/2 -5D/2 -7D/2 (b) 7D/2 5D/2 3D/2 D/2 -D/2 -3D/2 -5D/2 7/11/2017 10 Signals & Waves In the previous graph the vertical lines designate the number of times the function is sampled. The horizontal lines designate how many quantized levels are needed to represent the value of the amplitude accurately. Here 8 levels are needed corresponding to 3 bits (23=8). 7/11/2017 11 Signals & Waves 7/11/2017 12 Signals & Waves Sent Received • e.g. AM, FM, TV transmission Sent Received • e.g digital telephone, CD Audio 7/11/2017 13 Transmission segment Source 7/11/2017 Repeater Repeater Destination 14 Information 1 0 1 1 0 1 +1 Amplitude Shift Keying 0 T 2T 3T 4T 5T 6T 0 T 2T 3T 4T 5T 6T 0 T 2T 3T 4T 5T 6T -1 +1 Frequency Shift Keying -1 +1 Phase Shift Keying -1 7/11/2017 15 t Frequency (Hz) 102 10 10-2 10-6 x rays 10-4 gamma rays 1012 1014 1016 1018 1020 1022 1024 ultraviolet light broadcast radio 104 1010 visible light 108 infrared light 106 106 microwave radio 104 power & telephone 102 10-8 10-10 10-12 10-14 Wavelength (meters) 7/11/2017 16 Frequency (Hz) 104 105 106 108 107 109 1012 1011 1010 FM radio & TV Wireless cable AM radio Cellular & PCS satellite & terrestrial microwave LF 10 4 MF 103 HF 102 VHF 101 UHF 1 SHF 10-1 EHF 10-2 10-3 Wavelength (meters) 7/11/2017 17 Multiplexing (a) 7/11/2017 (b) A A A B B B C C C A Trunk group MUX MUX B C 18 (a) Individual signals occupy W Hz A f W 0 B 0 f W C f 0 W (b) Combined signal fits into channel bandwidth A B C f 7/11/2017 Frequency Division Multiplexing 19 Signals & Waves 7/11/2017 FDM was introduced in the telephone networks in the 1930s. Each voice signal requires 4kHz of bandwidth. The basic analog multiplexer combines 12 voice channels in one line. The multiplexer modulates each voice signal so that it occupies a 4kHz slot in the band between 60 and 108 kHz. A hierarchy of such groupings has been 20 Time Division Multiplexing (a) Each signal transmits 1 unit every 3T seconds A1 0T A2 6T 3T B1 B2 C1 0T t 6T 3T 0T t C2 t 6T 3T (b) Combined signal transmits 1 unit every T seconds A1 B1 0T 1T 2T 7/11/2017 C1 A2 3T 4T B2 C2 t 5T 6T 21 Signals & Waves 7/11/2017 TDM was introduced in the telephone network in early 1960s. The T-1 carrier system was designed to carry 24 digital connections. It uses a transmission frame that consists of 24 slots of 8 bits each. The beginning of each frame is indicated by a single bit that follows a periodic pattern. The speed is then (1+ 24*8)bits/frame * 8000frames/sec=1.544Mbps 22 100 50 10 Loss (dB/km) 5 Infrared absorption 1 0.5 Rayleigh scattering 0.1 0.05 0.01 0.8 1.0 1.2 1.4 1.6 1.8 Wavelength (m) 7/11/2017 23 Signals & Waves The previous Figure shows that a range of low attenuation wavelengths about 100nm wide is available in the 1300nm range. This range corresponds to a bandwidth of 18teraherz (THz). Another band of about 100nm in the 1550nm range provides another 19 THz of bandwidth. 7/11/2017 24 Signals & Waves Wavelength-division multiplexing (WDM) can be viewed as an optical domain version of FDM in which multiple information signals modulate optical signals at different optical wavelengths (colors). The resulting signals are combined and transmitted simultaneously over the same optical fiber. Prisms and diffraction gratings can be used to combine and split color signals (i.e, it is not necessary to change the optical signals to electrical signals first for switching purposes). 7/11/2017 25 Wavelength-division multiplexing (WDM) 1 2 m 7/11/2017 Optical MUX Optical deMUX 1 2 . m 1 2 Optical fiber m 26 Signals & Waves 7/11/2017 27 Signals & Waves 7/11/2017 28 Signals & Waves 7/11/2017 29 Signals & Waves 7/11/2017 30