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Bandwidth • Signal bandwidth: – We can divide signals into two categories: The pure tone signal (the sinusoidal wave, consisting of one frequency component), and complex signals that are composed of several components, or sinusoids of various frequencies. T=1x10-3 s 0 1 Pure signal f=1/1x10-3 =1000Hz=1 kHz t (ms) The bandwidth of a signal composed of components of various frequencies (complex signal) is the difference between its highest and lowest frequency components, and is expressed in Hertz (Hz) - the same as frequency. For example, a square wave may be constructed by adding sine waves of various frequencies: Pure tone 150 Hz sine wave Pure tone 450 Hz sine wave Approaching a 150 Hz square wave (ms) The resulting wave resembles a square wave. If more sine waves of other frequencies were added, the resulting waveform would more closely resemble a square wave Since the resulting wave contains 2 frequency components, its bandwidth is around 450-150=300 Hz. Channel bandwidth: – The bandwidth of a channel (medium) is defined to be the range of frequencies that the medium can support. Bandwidth is measured in Hz – With each transmission medium, there is a frequency range of electromagnetic waves that can be transmitted: Increasing bandwidth » Twisted pair cable: 0 to 109 Hz (Bandwidth : 109 Hz) » Coax cable: 0 to 1010 Hz (Bandwidth : 1010 Hz) » Optical fiber: 1014 to 1016 Hz (Bandwidth : 1016 -1014 = 9.9x1015 Hz) – Optical fibers have the highest bandwidth (they can support electromagnetic waves with very high frequencies, such as light waves) – The bandwidth of the channel dictates the information carrying capacity of the channel – This is calculated using Shannon’s channel capacity formula The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal. •Spectrum (range of frequencies) •Absolute bandwidth: width of spectrum •Bandwidth (width of the spectrum) •Two composite signals with different spectrums might have the same bandwidth. Signal A [10,20] vs. Signal B [90,100] The bandwidth of periodic and nonperiodic composite signals Example If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, what is its bandwidth? Draw the spectrum, assuming all components have a maximum amplitude of 10 V. Solution Let fh be the highest frequency, fl the lowest frequency, and B the bandwidth. Then The spectrum has only five spikes, at 100, 300, 500, 700, and 900 Hz Bandwidth for above example Example A non periodic composite signal has a bandwidth of 200 kHz, with a middle frequency of 140 kHz and peak amplitude of 20 V. The two extreme frequencies have an amplitude of 0. Draw the frequency domain of the signal. Solution Let fh be the highest frequency, and fl the lowest frequency. Then fh – fl = 200 (fh+fl)/2 = 140 The lowest frequency must be at 40 kHz and the highest at 240 kHz. Bandwidth for above example In networking, we use the term bandwidth in two contexts • The first, bandwidth in Hertz, refers to the range of frequencies in a composite signal or the range of frequencies that a channel can pass. •The second, bandwidth in bits per second, refers to the speed of bit transmission in a channel or link. Signal-to-Noise Ratio • S/N is normally measured in dB (decibel). It is a relationship between the signal we want versus the noise that we do not want, which is in the medium. • It can be thought of as a fractional relationship (that is, before we take the logarithm): Decibel: measures the relative strengths of two signals or one signal at two different points. dB 10 log 10 P2 P1 It is negative if a signal is attenuated and positive if a signal is amplified SNR = Average signal power/Average noise power SNRdB 10 log 10 SNR Example Consider an extremely noisy channel in which the value of the signal-to-noise ratio is almost zero. In other words, the noise is so strong that the signal is faint. For this channel the capacity C is calculated as This means that the capacity of this channel is zero regardless of the bandwidth. In other words, we cannot receive any data through this channel. Example We can calculate the theoretical highest bit rate of a regular telephone line. A telephone line normally has a bandwidth of 3000. The signal-to-noise ratio is usually 3162. For this channel the capacity is calculated as This means that the highest bit rate for a telephone line is 34.860 kbps. If we want to send data faster than this, we can either increase the bandwidth of the line or improve the signal-to-noise ratio. For practical purposes, when the SNR is very high, we can assume that SNR + 1 is almost the same as SNR. In these cases, the theoretical channel capacity can be simplified to For example, The signal-to-noise ratio is often given in decibels. Assume that SNRdB = 36 and the channel bandwidth is 2 MHz. Calculate theoretical channel capacity Digital Signals In addition to being represented by an analog signal, information can also be represented by a digital signal. For example, a 1 can be encoded as a positive voltage and a 0 as zero voltage. A digital signal can have more than two levels. In this case, we can send more than 1 bit for each level. Frequency Vs Bit Rate Frequency: The number of periods in 1 sec. Bit rate: The number of bits sent in 1sec, expressed in bits per second (bps). Bits per second (bps): The number of bits transmitted across a medium in a given second. Baud rate: The number of times a signal changes value per second. bps and baud rate are not always the same. Two digital signals: one with two levels and the other with four signal levels Example A digital signal has eight levels. How many bits are needed per level? We calculate the number of bits from the formula Each signal level is represented by 3 bits. Example What is the bit rate for high-definition TV (HDTV)? Solution HDTV uses digital signals to broadcast high quality video signals. The HDTV screen is normally a ratio of 16 : 9. There are 1920 by 1080 pixels per screen, and the screen is renewed 30 times per second. Twenty-four bits represents one color pixel. The TV stations reduce this rate to 20 to 40 Mbps through compression. Baseband A baseband is an adjective that describes signals and systems whose range of frequencies is measured from zero to a maximum bandwidth or highest signal frequency; it is sometimes used as a noun for a band of frequencies starting at zero. It can often be considered as synonym to low pass, and antonym to passband. Baseband transmission: sending a digital signal over a channel without changing the digital signal to an analog signal. •If we need to send bits faster, we need more bandwidth.