Transcript
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FDM vs TDMFrequency division multiplexing(Analog Exchange)
• 1 chl 300 Hz to 3400 Hz
• 3chls : Basic Group(BG): 12 – 24 KHz
• 12chls: Sub Group : 4BG: 60 -108KHz
• 60chls:SuperGroup:5SubGp: 312 – 552 KHz
• 900chls:Master Group:16 super Gp 312 – 4028 KHz
• 2700chls:Super Master Group: (3 MG):312 -12336 KHz
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Time division multiplexing
• Voice frequency 4KHZ
• Sampling rate 8kilo samples per second
• Each sample requires 8bits
• 1 voice frequency requires 64kbps
• 32 voice frequencies (PCM) requires 2Mbps
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PCM
Ist order
2nd order
3rd order
Higher order
2Mb
140Mb/STM1
34Mb
8Mb
2Gb/STM16
System Capacity
30 120 480 1920 30000
No. of voice channels
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Chapter 2Discrete-Time signals and systems
• Representation of discrete-time signals
• (a)Functional
• (b)Tabular
• (c)Sequence
• Examples of singularity functions
Impulse, Step and Ramp functions and shifted Impulse, Step and Ramp functions
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Energy and power signals
• If E=infinity and P= finite then the signal is power signal
• If E=finite and P= zero then the signal is Energy signal
• If E=infinite and P= infinite then the signal is neither energy nor power signal
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Periodic and non-periodic signals
• X(n+N) = x(n) for all n, and N is the fundamental time period.
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Block diagram Representation of Discrete-Time systems
• Adder
• A constant multiplier
• A signal multiplier
• A unit delay element
• A unit advance element
• Folding
• Modulator
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Classcification of Discrete-time systems
• Static vs dynamic systems
• Time invariant vs time-variant systems
• Linear vs nonlinear systems systems
• Causal vs non causal systems
• Stable vs unstable systems
• Recursive vs non recursive systems
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Static vs dynamic systems
• A discrete time system is called static or memoryless if its output at any instant ‘n’ depends at most on the input sample at the same time, but not on past or future samples of the input. In any other case, the system is said to be dynamic or to have memory.
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Time invariant vs time-variant systems
• A system is called time-invariant if its impute-output characteristics do not change with time.
• X(n-k) y(n-k)
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Linear vs nonlinear systems systems
• Linear system obeys the principle of superposition
• T[a x1(n) + b x2(n)] = a T[x1(n)] + b T[x2(n)]
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LTI system
• The system which obeys both Linear property and Time invariant property.
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Causal vs non causal systems
• A system is said to be causal if the output of the system at any time depends only on present and past inputs, but not depend on future inputs. If a system doesnot satisfy this definition , it is called noncausal system.
• All static systems are causal systems.
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Stable vs unstable systems
• An arbitrary relaxed system is said to be bounded input – bounded output (BIBO) stable if an only if every bounded input produces a bounded output.
• y(n) = y2(n) + x(n) where x(n)=δ(n)
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Recursive vs non recursive systems
• Recursive system uses feed back
• The output depends on past values of output.
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Interconnection of discrete time systems
• Addition/subtraction
• Multiplication
• Convolution
• Correlation (a) Auto (b) Cross correlation
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Convolution
• Used for filtering of the signals in time domain. Used for LTI systems only
• 1.comutative law
• 2.associative law
• 3.d17istributive law
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Correlation
• Auto correlation improves the signal to noise ratio
• Cross correlation attenuates the unwanted signal
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Digital signal Distortion
Distortion less transmission y(n) = α.x(n-l)
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Random signal vs Deterministic signal
• Deterministic signal: can be represented in mathematical form since the present, past and future values can be predicted based on the equation
• Random signal: can not be put in mathematical form. Only the average, rms, peak value and bandwidth can be estimated with in a given period of time.
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Advantages of digital over analog signals
• Faithful reproduction by reshaping
• Information is only 2 bits (binary)
• Processed in microprocessor
• Signals can be compressed
• Error detection and correction available
• Signals can be encrypted and decrypted
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Disadvantages
• Occupies more bandwidth• Difficult to process the digital signals in
microwave frequency range since the speed of the microprocessor is limited (3 GHz)
• Digital signals have to be converted into analog signals for radio communication.
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Advantages of Digital signal processing over analog signal processing
• Flexible in system reconfiguration• Accuracy , precise and better tolerances• Storage • Low cost• Miniaturization• Single micro processor• Software operated (programmed)• Artificial intelligence
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Multi channel and Multidimensional Signals
• Multichannel:
• S(t) = [ s1(t), s2(t), s3(t)…]
• Multi Dimensional: A value of a signal is a function of M independent variables.
• S(x,y,t) = [ s1(x,y,t), s2(x,y,t), s3(x,y,t)…]
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Analog-discrete-digital-PCM
• Analog frequency
• Demo of Time and frequency signals
• Digital frequency
• Periodic and aperiodic signals
• Sampling theorem
• A to D / D to A conversion
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Variable sampling rate
28Ch2. Formatting [6]
t
x(t)
0-f m f m
f
|X(f)|
0
(a) (b)
. . . . . .
4T s2T s-2T s 0-4T s
t
x δ (t) = n = -∞
∞δ (t-nT s) X δ (f) =
n = -∞
∞δ (f-nf s)
. . .. . .
-2f s 2f sfs-f s 0f
1T s
(d)(c)
4T s2T s-2T s 0-4T s
(e)
. . .. . .
-2f s 2f sfs-f s 0f
(f)
-f m fm
|X s(f)|x s(t) = x(t) x δ (t)
t
[Fig 2.6] Sampling theorem using the frequency convolution property of the Fourier transform
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Sampling
• The operation of A/D is to generate sequence by taking values of a signal at specified instants of time
• Consider a system involving A/D as shown below:
)(tx Analog-to-digital
converter
Digital-to-analog
converter
)(nTx
][nx
)(txh
t
xh(t)x (t)
5T4T3T2TT0
x(t) is the input signal and xh(t) is the sampled and reconstructed signal
T = sample period
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Impulse Sampling
What is Impulse Sampling?
– Suppose a continuous–time signal is given by x(t), - < t < +
– Choose a sampling interval T and read off the value of x(t) at times nT, n = -,…,-1,0,1,2,…,
– The values x(nT) are the sampled version of x(t)
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Impulse Sampling
• The sampling operation can be represented in a block diagram as below:
• This is done by multiplying the signal x(t) by a train of impulse function T(t)
• The sampled signal here is represented by xS(t) and the sampling period is T
)(tx )(txS
nT nTtt )()(
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Impulse Sampling
• From the block diagram, define a mathematical representation of the sampled signal using a train of -function
ttx
nTttx
nTtnTxtx
T
n
ns
continuous –time signal function
Train of periodic impulse function
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Impulse Sampling
• We therefore have
• And
It turns out that the problem is much easier to understand in the frequency domain. Hence, we determine the Fourier Transform of xs(t).
n
tjnT e
Tt 0
1
n
tjn
n
tjns
etxT
eT
txtx
0
0
1
1
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Impulse Sampling
• Looking at each term of the summation, we have the frequency shift theorem:
• Hence the Fourier transform of the sum is: 0
0 nXetxCTFT
tjn
n
s nXT
X 0
1
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Impulse Sampling
i.e. The Fourier transform of the sampled signal is simply the Fourier transform of the continuous signal repeated at the multiple of the sampling frequency and scaled by 1/T.
1
0
X(
1/T
0
Xs(
-0 0-20 20
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Nyquist Sampling Theorem
• The Nyquist sampling theorem can be stated as:
If a signal x(t) has a maximum frequency content (or bandwidth) max, then it is possible to reconstruct x(t) perfectly from its sampled version xs(t) provided the sampling frequency is at least 2 max
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Nyquist Sampling Theorem
• The minimum sampling frequency of 2 max is known as the Nyquist frequency, Nyq
• The repetition of X() in the sampled spectrum are known as aliasing. Aliasing will occur any time the sample rate is not at least twice as fast as any of the frequencies in the signal being sampled.
• When a signal is sampled at a rate less than Nyq the distortion due to the overlapping spectra is called aliasing distortion
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Multichannel Tx-Rx
• Station A cable Station B
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Design of discrete time systems
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PERIODIC AND APERIODIC SIGNALS OF ANALOG AND DISCRETE
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Computing the Z-transform: an example
• Example 1: Consider the time function
x[n] nu[n]
X(z) x[n]z n nz n (z 1)n
n0
n0
n
1
1 z 1 z
z
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Another example …
•
x[n] nu[ n 1]
X(z) x[n]z n nz n (z 1)n
n
1
n
1
n
l n;n l ;n 1 l 1
(z 1)n (z 1)l 1 l1
n
1
(z 1)l 1 1
1 z 1 1
1 z 1l0
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Frequency domain of a rectangular pulse
Fourier coefficients of the rectangular pulse train with time period Tp is fixed and the pulse width tow varies.
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SAMPLING IN TIME DOMAIN
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