Signal and System

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• Signal:

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A signal is defined as a function of one or more variables which conveys information on the nature of a physical phenomenon. The value of the function can be a real valued scalar quantity, a complex valued quantity, or perhaps a vector.

• System:A system is defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

• Continuos-Time Signal:

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A signal x(t) is said to be a continuous time signal if it is defined for all time t.

• Discrete-Time Signal:A discrete time signal x[nT] has values specified only at discrete points in time.

• Signal Processing:A system characterized by the type of operation that it performs on the signal. For example, if the operation is linear, the system is called linear. If the operation is non-linear, the system is said to be non-linear, and so forth. Such operations are usually referred to as “Signal Processing”.

Basic Elements of a Signal Processing System

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Analog Signal Processor

Analog inputsignal

Analog outputsignal

Analog Signal Processing

Digital Signal Processor

A/Dconverter

D/Aconverter

Digital Signal Processing

Analog inputsignal

Analog outputsignal

• Advantages of Digital over Analogue Signal Processing:

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A digital programmable system allows flexibility in reconfiguring the DSP operations simply by changing the program. Reconfiguration of an analogue system usually implies a redesign of hardware, testing and verification that it operates properly.

DSP provides better control of accuracy requirements.

Digital signals are easily stored on magnetic media (tape or disk).

The DSP allows for the implementation of more sophisticated signal processing algorithms.

In some cases a digital implementation of the signal processing system is cheaper than its analogue counterpart.

DSP Applications

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Space Space photograph enhancementData compressionIntelligent sensory analysis

MedicalDiagnostic imaging (MRI, CT, ultrasound, etc.)Electrocardiogram analysisMedical image storage and retrieval

Image and sound compression for multimedia presentation.Movie special effectsVideo conference calling

Commercial

Video and data compressionecho reductionsignal multiplexingfiltering

Telephone

DSP Applications (cont.)

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Military

RadarSonarOrdnance GuidanceSecure communication

IndustrialOil and mineral prospectingProcess monitoring and controlNon-destructive testingCAD and design tools

Scientific

Earth quick recording and analysisData acquisitionSpectral AnalysisSimulation and Modeling

Classification of Signals

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•Deterministic SignalsA deterministic signal behaves in a fixed known way with respect to time. Thus, it can be modeled by a known function of time t for continuous time signals, or a known function of a sampler number n, and sampling spacing T for discrete time signals.

• Random or Stochastic Signals:In many practical situations, there are signals that either cannot be described to any reasonable degree of accuracy by explicit mathematical formulas, or such a description is too complicated to be of any practical use. The lack of such a relationship implies that such signals evolve in time in an unpredictable manner. We refer to these signals as random.

Even and Odd Signals

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A continuous time signal x(t) is said to an even signal if it satisfies the conditionx(-t) = x(t) for all t

The signal x(t) is said to be an odd signal if it satisfies the conditionx(-t) = -x(t)

In other words, even signals are symmetric about the vertical axis or time origin, whereas odd signals are antisymmetric about the time origin. Similar remarks apply to discrete-time signals.

Example:

even

odd odd

Tutorial 1: Q1Define xe(t) to be even and xo(t) to be odd part of a signal x(t). Show that

Q2:Decompose the following signal into its even and odd parts:

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)t(x)t(x21tx

)t(x)t(x21)t(x

0

e

Note: The above definitions of even and odd signals assume that the signals are real valued. A complex valued signal x(t) is said to

to be conjugate symmetric if it satisfies the condition x(-t) = x*(t)

-1 1

x(t)

Periodic Signals

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A continuous signal x(t) is periodic if and only if there exists a T > 0 such thatx(t + T) = x(t)where T is the period of the signal in units of time.

f = 1/T is the frequency of the signal in Hz. W = 2/T is the angular frequency in radians per second.

The discrete time signal x[nT] is periodic if and only if there exists an N > 0 such thatx[nT + N] = x[nT]where N is the period of the signal in number of sample spacings.

Example:

0 0.2 0.4

Frequency = 5 Hz or 10 rad/s

Continuous Time Sinusoidal Signals

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A simple harmonic oscillation is mathematically described asx(t) = Acos(wt + )

This signal is completely characterized by three parameters:A = amplitude, w = 2f = frequency in rad/s, and = phase in radians.

A T=1/f

Discrete Time Sinusoidal Signals

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A discrete time sinusoidal signal may be expressed as x[n] = Acos(wn + ) - < n <

Properties: • A discrete time sinusoid is periodic only if its frequency is a rational number.

• Discrete time sinusoids whose frequencies are separated by an integer multiple of 2 are identical.

• The highest rate of oscillation in a discrete time sinusoid is attained when w = ( or w = - ), or equivalently f = 1/2 (or f = -1/2).

0 2 4 6 8 10-1

0

1

Energy and Power Signals

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The total energy of a continuous time signal x(t) is defined as

dttxdt)t(xlimE 2T

T

2

Tx

And its average power is

2/T

2/T

2Tx dt)t(x

T1limP

In the case of a discrete time signal x[nT], the total energy of the signal is

n

2dx nxTE

And its average power is defined by

2N

NnNdx nTx1N2

1limP

Energy and Power Signals

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• A signal is referred to as an energy signal, if and only if the total energy of the signal satisfies the condition0 < E <

•On the other hand, it is referred to as a power signal, if and only if the average power of the signal satisfies the condition0 < P <

•An energy signal has zero average power, whereas a power signal has infinite energy.

•Periodic signals and random signals are usually viewed as power signals, whereas signals that are both deterministic and non-periodic are energy signals.

Example1:Compute the signal energy and signal power forx[nT] = (-0.5)nu(nT), T = 0.01 seconds

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Solution:

2

0n

n2N

NnNdx 5.001.0nTxTlimE

0n

nn2

0n25.001.05.001.0

.......25.025.025.0101.0 32

75/125.0101.0

Since Edx is finite, the signal power is zero.

Example2:Repeat Example1 for y[nT] = 2ej3nu[nT], T = 0.2 second.

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Solution:

N

0n

2n3j

N

2N

NnNdx e21N2

1lim)nT(y1N2

1limP

1N2)1N(4lim1

1N24lim2

1N21lim

N

0n NN

N

0n

2

N

2214

1N21

1N2N4lim

N

What is energy of this signal?

Tutorial 1: Q3

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Determine the signal energy and signal power for each of the given signals and indicate whether it is an energy signal or a power signal?

],3n[u)2.0(3]nT[y n (a) T = 2 ms

(b) ]1n[u1.14]nT[z n T = 0.02 s

Basic Operations on Signals(a) Operations performed on dependent variables

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1. Amplitude Scaling:let x(t) denote a continuous time signal. The signal y(t) resulting from amplitude

scaling applied to x(t) is defined byy(t) = cx(t)where c is the scale factor.

In a similar manner to the above equation, for discrete time signals we writey[nT] = cx[nT]

x(t)

2x(t)

2. Addition:Let x1 [n] and x2[n] denote a pair of discrete time signals. The signal y[n] obtained by the addition of x1[n] + x2[n]

is defined asy[n] = x1[n] + x2[n]Example: audio mixer

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3. Multiplication:Let x1[n] and x2[n] denote a pair of discrete-time signals. The signal y[n] resulting

from the multiplication of the x1[n] and x2[n] is defined byy[n] = x1[n].x2[n]

Example: AM Radio Signal

(b) Operations performed on independent variable• Time Scaling:

Let y(t) is a compressed version of x(t). The signal y(t) obtained by scaling the independent variable, time t, by a factor k is defined byy(t) = x(kt)

• if k > 1, the signal y(t) is a compressed version of x(t).• If, on the other hand, 0 < k < 1, the signal y(t) is an

expanded (stretched) version of x(t).

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Example of time scaling

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

exp(-2t)

exp(-t)

exp(-0.5t)

Expansion and compression of the signal e-t.

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Time scaling of discrete time systems

Time Reversal• This operation reflects the signal about t = 0 and thus reverses the signal

on the time scale.

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Time ShiftA signal may be shifted in time by replacing the independent variable n by n-

k, where k is an integer. If k is a positive integer, the time shift results in a delay of the signal by k units of time. If k is a negative integer, the time shift results in an advance of the signal by |k| units in time.

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Analog to Digital and Digital to Analog Conversion

• A/D conversion can be viewed as a three step process

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1. Sampling: This is the conversion of a continuous time signal into a discrete time signal obtained by taking “samples” of the continuous time signal at discrete time instants. Thus, if x(t) is the input to the sampler, the output is x(nT), where T is called the Sampling interval.

2. Quantization: This is the conversion of discrete time continuous valued signal into a discrete-time discrete-value (digital) signal. The value of each signal sample is represented by a value selected from a finite set of possible values. The difference between unquantized sample and the quantized output is called the Quantization error.

Analog to Digital and Digital to Analog Conversion (cont.)

3. Coding: In the coding process, each discrete value is represented by a b-bit binary sequence.

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QuantizerSampler Coderx(t) 0101...

A/D Converter