Baseand Data Transmission Data are sent without using a carrier signal Example: Bipolar NRZ (non-return-to-zero signaling is represented y is represented y T A -A T : it duration is represented y BT.
Passand Data Transmission Data are sent using a carrier signal Example: PSK (Phase Shift Keying is represented y T T : it duration is represented y is represented y BT.
Optimum receiver: matched filter A asic prolem in communication is detecting a signal transmitted over a channel that is corrupted y channel noise. Transmitted signal g(t Received signal x ( t g( t + w( t BT.3
Matched Filter A matched filter is a linear filter designed to provide the maximum signal-to-noise power ratio at its output. This is very often used at the receiver. It is assumed that the receiver has knowledge of the waveform of the signal g(t. Channel Matched filter Sampler BT.4
Matched Filter Signal Power Let G(f and H(f denoted the Fourier Transform of g(t and h(t, we have G o ( g f H ( ( t f G( f H ( f G( f exp( jπft df (Inverse FT The signal power at T g T H ( f G( f exp( jπt ( df BT.5
Matched Filter Noise Power Since w(t is white with a power spectral density N /, the spectral density function of Noise is The noise power N S N ( f f H ( N E[ n ( t] H ( f df BT.6
Matched Filter S/N Ratio Thus the signal to noise ratio ecome η H ( f G( f exp( jπft df N H ( f df....( Our prolem is to find, for a given G(f, the particular form of the transfer function H(f of the filter that makes η at maximum. BT.7
Matched Filter Schwarz s inequality: If φ ( x dx < and φ ( x dx φ ( x φ ( x dx φ( x dx φ ( x < dx * This equality holds if and only if, we have φ ( x kφ ( x where k is an aritrary constant, and * denotes complex conjugation. BT.8
Matched Filter jπft Setting φ( t H ( f and φ ( t G( f e and then applying the Schwarz s inequality to equation (, we have η η H ( f G( N N o H ( f o f exp( jπft df df H ( f G( H ( f df f df df j ft ( e π φ ( x φ ( x dx φ( x dx φ ( x BT.9 dx
Matched Filter Therefore, η N G( f df or η E N (3 where E G( f df is the input signal energy BT.
and G * ( Matched Filter The maximum S/N ratio is then otained when * * H ( f kg ( f exp( jπft φ ( x kφ ( x Taking the inverse Fourier transform of H(f we have * h ( t k G ( f exp[ jπf ( T t] df f h( t G( f for real signal g(t k G( f exp[ jπf ( T t] df kg( T t It shows that the impulse response of the filter is the timereversed and delayed version of the input signal g(t. BT.
Matched Filter Example: The signal is a rectangular pulse. g(t g( t A A T t -T t The impulse response of the matched filter has exactly the same waveform as the signal. h(t ka g( T t T t BT.
Matched Filter The output signal of the matched filter has a triangular waveform. g o (t g ( t g( t h( t ka o T T t In this special case, the matched filter can e implemented using a circuit known as integrate-anddump circuit. r(t T Sample at t T BT.3
BT.4 Realization of the Matched filter The matched filter output is ( ( ] ( [ ( ( ( ( ( ( t T g t h d t T g r d t h r t h t r t y t t Q τ τ τ τ τ τ t d g r T y ( ( ( τ τ τ T r(t g(t Correlator
Error Rate (Binary PAM Signaling Consider a non-return-to-zero (NRZ signaling (sometime called ipolar. Symol and are represented y positive and negative rectangular pulses of equal amplitude and equal duration. Noise The channel noise is modeled as additive white Gaussian (AWG noise of zero mean and power spectral density N o /. In the signaling interval t T, the received signal is x( t + A + A + w( t w( t symol was sent symol was sent A T A is the transmitted pulse amplitude T is the it duration BT.5 t
Error Rate (Binary PAM Receiver x(t T T y Decision device if if y y > λ < λ Sample at t T Matched filter λ It is assumed that the receiver has prior knowledge of the pulse shape, ut not its polarity. Given the noisy signal x(t, the receiver is required to make a decision in each signaling interval. BT.6
Error Rate (Binary PAM In actual transmission, a decision device is used to determine the received signal. There are two types of error Case : Symol is chosen when a was actually transmitted Case : Symol is chosen when a was actually transmitted Example: BSC p p p p BT.7
Error Rate (Binary PAM Case I Suppose that a symol was sent then the received signal is x(t -A + n(t + The matched filter output y(t is y( t T T x( t dt A + T T n( t dt BT.8
BT.9 Error Rate (Binary PAM As the noise is white and Gaussian, y(t is also Gaussian with the following parameters: mean of y(t: A dt t n T E A E dt t n T A E y E y T T + + ] ( [ ] [ ] ( [ (
Error Rate (Binary PAM Variance of y(t: σ y E[( y y ] N T o (Proof refers to p.54, S. Haykin, Communication Systems -A BT.
Error Rate (Binary PAM The proaility density function of a Gaussian distriuted signal is f y ( y y ( y exp( πσ σ y Therefore, the conditional proaility density function of the random variale y, given that was sent, is y f y ( y πn / T exp( ( y + A N / T BT.
Error Rate (Binary PAM Let p denote the conditional proaility of error, given that symol was sent This proaility is defined y the shaded area under the curve of f y (y from the threshold λ to infinity, which corresponds to the range of values assumed y y for a decision in favor of symol p P( y > λ BT.
Error Rate (Binary PAM The proaility of error is P P( y > λ Symol was sent λ f y πn ( y dy / T λ exp( ( y + A N / T dy Assuming that symols and occur with equal proaility, i.e. P P / If there is no noise, the output at the matched filter will e A for symol and A for symol. The threshold λ is set to e. BT.3
Error Rate (Binary PAM P πn / T λ exp( ( y + A N / T dy y + A N Define a new variale z and then dy dz. N o / T T P exp( z π E / N o dz E where is the transmitted signal energy per it, defined y E A T BT.4
Error Rate (Binary PAM At this point we find it convenient to introduce the definite integration called complementary error function. erfc( u exp( z dz π u Therefore, the conditional proaility of error P erfc( E N ( Note: erf( u exp( z π u dz and erfc(u-erf(u BT.5
Error Rate (Binary PAM In some literature, Q function is used instead of erfc function. u Q( x exp( du π x Q( x erfc( and erfc( x Q( x x BT.6
Error Rate (Binary PAM Case II Similary, the conditional proaility density function of y given that symol was sent, is ( y A f y ( y exp( π N / T N / T P πn / T λ exp( ( y N A / T dy BT.7
By setting λ and putting y A z N / T we find that P P Error Rate (Binary PAM The average proaility of symol error P e is otained as P e P P + P P If the proaility of and are equal and equal to ½ E Pe erfc( N BT.8
Error Rate (Binary PAM BT.9
Example A polar trinary waveform in additive Gaussian noise with proaility density function ( P o (y, P (y and P (y as shown elow. Calculate the net proaility of error if the decision thresholds are set at ±A/4. P ( y P ( y P ( y -A/ -A/4 A/4 +A/ BT.3
Example As the mean of symol is y A / and the variance is N σ T ( y + A/ Pe exp( dy πn / T N / T A / 4 error Put z we have y + A/ N o / T P e π E erfc / 4N o E exp( z o 4N o dz -A/ -A/4 where E A T BT.3
BT.3 Example Similarly, + + + o o o N E z N E z A A e N E N E N E dz e dz e dy T N y dy T N y T N P o o 6 erfc 6 erfc 6 erfc / exp( / exp( / /6 /6 4 / 4 / π π error A/4 -A/4
Example By symmetry, Pe Pe erfc 4 ( E / N If P(P(P(/3, we have o P e P 3 e P + P ( e ( e [ erfc( E / 4N + erfc( E /6N ] P + o P P( o BT.33
Example Trinary PAM P e Binary PAM E / N o (db BT.34
Intersymol interference (ISI (ISI it is a signal-dependent form of interference that arises ecause of deviations in the frequency response of a channel from the ideal channel. Example: Bandlimited channel Time Domain Bandlimited channel Frequency domain BT.35
Intersymol interference (ISI This non-ideal communication channel is also called dispersive channel The result of these deviation is that the received pulse corresponding to a particular data symol is affected y the previous symols and susequent symols. BT.36
Example Waveform of t t BT.37
Two scenarios Intersymol interference (ISI I. The effect of ISI is negligile in comparison to that of channel noise. use a matched filter, which is the optimum linear timeinvariant filter for maximizing the peak pulse signalto-noise ratio. II. The received S/N ratio is high enough to ignore the effect of channel noise (For example, a telephone system control the shape of the received pulse. BT.38
ISI Consider a inary system, the incoming inary sequence { } k consists of symols and, each of duration T. The pulse amplitude modulator modifies this inary sequence into a new sequence of short pulses (approximating a unit impulse, whose amplitude a k is represented in the polar form + if k ak if k { } k Pulseamplitude modulator { } k a Transmit s(t filter g(t Channel h(t x o (t x(t w(t White noise BT.39
ISI Example:{ k } k a δ ( t k kt { }: a k T t BT.4
ISI The short pulses are applied to a transmit filter of impulse response g(t, producing the transmitted signal s ( t a g( t k k kt The signal s(t is modified as a result of transmission through the channel of impulse response h(t. In addition, the channel adds random noise to the signal. x( t ak g( t kt h( t + n( t { } k Pulseamplitude modulator k { } k a Transmit s(t filter g(t Channel h(t x o (t w(t x(t White noise BT.4
ISI The noisy signal x(t is then passed through a receive filter of impulse response c (t.the resulting output y(t is sampled and reconstruced y means of a decision device. x(t Receive filter c(t The receiver output is y(t Sample at t i it Decision device λ if if y( t µ ak p( t kt + n( t k y > λ y < λ where µ p( t g( t h( t c( t and µ is a constant. BT.4
Example: { } k ISI { }: a k aδ ( t a δ ( t T y(t T t assume n( t µ a p( t µa p( t T t y( t µ a p( t k kt + n( t k BT.43
BT.44 ISI The sampled output is : contriution of the i th transmitted it. : The residual effect of all other transmitted its. (This effect is called intersymol interference ( ] [( ( ] [( ( i i k k k i i k k i t n T k i p a a t n T k i p a t y + + + µ µ µ µa i i k k k T k i p a ] [( µ
Example: { } y(t k T ISI y ( ti µ a + µ i ak p[( i k T ] + n( ti k k i assume n( t µa µa p( T ( i, k t y( t µ a p( t k kt + n( t k µa3 p( T ( i, k 3 t t (i.e. i BT.45
Distortionless Transmission In a digital transmission system, the frequency response of the channel h(t is specified. We need to determine the frequency responses of the transmit g (t and receive filter c(t so as to reconstruct the original inary data sequence { k }. { } k Pulseamplitude modulator { } k a Transmit s(t filter g(t Channel h(t x o (t w(t x(t White noise x(t Receive filter c(t y(t Decision device if if y > λ y < λ Sample at t i it λ BT.46
BT.47 Distortionless Transmission The decoding requires that k i k i kt it p ( Ignore the noise ( ] [( ( ] [( ( i i k k k i i k k i t n T k i p a a t n T k i p a t y + + + µ µ µ y(t t ( assume t n y(t i µa i
Distortionless Transmission It can e shown that the condition i k p( it kt i k is equivalent to n P ( f n / T T BT.48
Example t p(t t p( p( T Sample points BT.49
Example p( f T sinc p(t /T f T T p( f p f / T p f / T ( ( n p ( f n / T T f BT.5
The simplest way of satisfying Ideal Nyquist Channel n P ( f n / T T is a rectangular function: W < f < p( f W f > W W / T W / W W / T BT.5
Ideal Nyquist Channel p( t sin(πwt πwt The special value of the it rate R / T W is called the Nyquist rate, and W is called the Nyquist andwidth. This ideal aseand pulse system is called the ideal Nyquist channel BT.5
Example Sampling instants BT.53
Ideal Nyquist Channel In practical situation, it is not easy to achieve it due to The system characteristics of P(f e flat from -/T up to /T and zero elsewhere. This is physically unrealizale ecause of the transitions at the edges. The function decreases as / t for large t, resulting in a slow rate of decay. Therefore, there is practically no margin of error in sampling times in the receiver. BT.54
BT.55 Raised Cosine Spectrum We may overcome the practical difficulties encountered y increasing the andwidth of the filter. Instead of using we use > < < W f W f W W f p ( > < < + + + W f W f W W W f P W f p f p ( ( ( W T /
Raised Cosine Spectrum A particular form is a raised cosine filter BT.56
BT.57 Raised Cosine Spectrum The frequency characteristic consists of a flat amplitude portion and a roll-off portion that has a sinusoidal form. The pulse spectrum p(f is specified in terms of a roll off factor α as follows: The frequency parameter and andwidth W are related y > < < ( sin 4 ( f W f f W f f f W W f W f f W f p π f f /W α
Raised Cosine Spectrum where α is the rolloff factor. It indicates the excess andwidth over the ideal solution (Nyquist channel where W/T. The transmission andwidth is ( + α W BT.58
Raised Cosine Spectrum The frequency response of α at,.5 and are shown in graph elow. We oserved that α at and.5, the function P(f cutoff gradually as compared with the ideal Nyquist channel and is therefore easier to implement in practice. BT.59
Raised Cosine Spectrum The time response p(t is otained as ( cos(παwt p t (sin c(wt( 6α W t The function p(t consists of two parts. The first part is a sinc function that is exactly as Nyquist condition ut the second part is depended on α. The tails is reduced if α is approaching. Thus, it is insensitive to sampling time errors. BT.6
BT.6
BT.6 Example For α, (f the system is known as the full-cosine rolloff characteristic. > < < + W f W f W f W f p cos 4 ( π
Example p( t sinc(wt 6W t BT.63
Example This time response exhiits two interesting properties: At t ±T / ± /4W we have p(t.5; that is, the pulse width measured at half amplitude is exactly equal to the it duration T. t T / BT.64
Example There are zero crossings at t ± 3T /, ± 5T /,... in addition to the usual crossings at the sampling times t ± T /, ± T /,... t 3T / t 5T / BT.65
Example These two properties are extremely useful in extracting a timing signal from the received signal for the purpose of synchronization. However, the price paid for this desirale property is the use of a channel andwidth doule that required for the ideal Nyquist channel corresponding to α. BT.66
Example What is the minimum andwidth for transmitting data at a rate of 33.6 kps without ISI? Answer: The minimum andwidth is equal to the Nyquist andwidth. Therefore, (BW min W R / 33.6/ 6.8 khz Note: If a % roll-off characteristic is used, andwidth W(+α 33.6 khz BT.67
Example Bandwidth requirement of the T system T system multiplex 4 voice inputs, ased on an 8-it PCM word. andwidth of each voice input (B 3. khz For converting the voice signal into inary sequence, The minimum sampling rate B 6. khz Sampling rate used in telephone system 8 khz BT.68
Example With a sampling rate of 8 khz, each frame of the multiplexed signal occupies a period of 5µs. : : No. of its 8 4+93 8 it from st input 8 it from nd input. 8 it from 4th input it for Synchronization 5 µs BT.69
Example Correspondingly, the it duration is 5 µs/93.647 µs. For eliminating ISI, the minimum transmission andwidth is / T 77kHz BT.7
Eye diagrams This is a simple way to give a measure of how severe the ISI (as well as noise is. This pattern is generated y overlapping the incoming signal elements. Example: ipolar NRZ PAM T BT.7
Eye diagrams Eye pattern is often used to monitoring the performance of aseand signal. The est time to sample the received waveform is when the eye opening is largest. Effects of noise are ignored BT.7
Eye diagrams The maximum distortion and ISI are indicated y the vertical width of the two ranches at sampling time. BT.73
Eye diagrams The noise margin or immunity to noise is proportional to the width of the eye opening. 3 BT.74
Eye diagrams The sensitivity of the system to timing errors is determined y the rate of closure of the eye as the sampling time is varied. 4 BT.75
Equalization In preceding sections, raised-cosine filters were used to eliminate ISI. In many systems, however, either the channel characteristics are not known or they vary. Example The characteristics of a telephone channel may vary as a function of a particular connection and line used. It is advantageous in such systems to include a filter that can e adjusted to compensate for imperfect channel transmission characteristics, these filters are called equalizers. BT.76
Before equalization After equalization BT.77
Transversal filter (zero-forcing equalizer x k BT.78
Equalization The prolem of minimizing ISI is simplified y considering only those signals at correct sample times. The sampled input to the transversal equalizer is x ( kt x k The output is y ( kt y k x x For zero ISI, we require that k y k k (* x BT.79
The output can e expressed as an x k N y k N nn a n x k n x k N k N a x k a N x k + N There are N+ independent equations in terms of a n. This limits us to N+ constraints, and therefore (* must e modified to y k k k ±, ±,..., ± N BT.8
BT.8 Equalization The N+ equations ecomes + + + M M M M L L L L M M L L M M L L L L N N N N N N N N N N N N N N N N N N N N o a a a a a x x x x x x x x x x x x x x x x x x x x x x x x x
Example Determine the tap weights of a three-tap, zero-forcing equalizer for the input where x., x., x., x.3, x., x for k > k N The three equations are a +.a.3a + a.a.3a +.a + a Solving, we otain a.779, a.847, a.8897 BT.8
Equalization The values of the equalized pulse are y., y.356, y y 3., y.36, y., y 3.,.85 This pulse has the desired zeros to either side of the peak, ut ISI has een introduced at sample points farther from the peak. BT.83
BT.84
Duoinary Signaling Intersymol interference is an undesirale phenomenon that produces a degradation in system performance. However, y adding intersymol interference to the transmitted signal in a controlled manner, it is possile to achieve a signaling rate equal to the Nyquist rate of W symols per second in a channel of andwidth W Hz. BT.85
Coding and decoding Consider a inary input sequence { k } consisting of uncorrelated inary symols and, each having duration T. This sequence is applied to a pulseamplitude modulator producing a two-level sequence of short pulses (approximating a unit impulse, whose amplitude is a k if if symol symol k k is is BT.86
This sequence is applied to a duoinary encoder as shown elow: { } a k { } c k Nyquist channel Delay T c k ak + ak One of the effects of the duoinary encoding is to change the input sequence { a k } of uncorrelated twolevel pulses into a sequence { c k } of correlated threelevel pulses. This correlation etween the adjacent pulses may e viewed as introducing intersymol interference into the transmitted signal in an artificial manner. BT.87
Consider { } Example k where the first it is a startup it. Encoding: {} k : { a k }: - - + - + + - c : - + {} k { } a k T t { } c k t BT.88
Decoding: Using the equation a k ck ak, or simply using If c + k, decide that a + k. If c k, decide that a k. If c k, decide opposite of the previous decision. BT.89
Duoinary Signaling: Impulse response and frequency spectrum Let us now examine an equivalent model of the duoinary encoder. The Fourier transfer of a delay can e descried as e πft, therefore, the transfer function of the encoder is H I ( f is jπft H ( f + e I The transfer function of the Nyquist channel is f < / T H N ( f otherwise BT.9
The overall equivalent transfer function H ( f is then given y H ( f H ( f H ( f for f < / T ( e I ( + e e jπft jπft N jπft + e jπft cosπft e jπft of the H(f has a gradual roll-off to the and edge which can e easily implemented BT.9
BT.9 The corresponding impulse response h(t is found y taking the inverse Fourier transform of H(f ( / sin( / ( / sin( / / sin( / ( / ( sin( / / sin( ( T t t T t T T T t T t T t T t T T t T T t T t T t t h + + π π π π π π π π π π
Notice that there are only two nonzero samples, at T -second intervals, give rise to controlled ISI from the adjacent it. The introduced ISI is eliminated y use of the decoding procedure. BT.93
M-ary PAM BT.94