Digital Signal Processing

Discussion #3Signals Operations

Tarun ChoubisaDept. of ETC,

KIIT University

18 January 2011 1

Classification: Causal/non-causal/anti-causal • A system for which the output at any instant depends only on the past or/and present values

of the input( not on future samples) is called as causal system. Referred to as non-anticipative, as the system output does not anticipate future values of the input

• E.g. y(n)= n*x(n) , y(n)=x(n) +x(n-1)

• All real-time physical systems are causal, because time only moves forward.

• Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast.

• Fact: – A causal system may be memory or memory-less system.

– Any memoryless system is causal.

– The composition of causal systems is causal

• A system for which the output at any instant depends also on future values (in addition to possible dependence on past or current input values)of the input , is called as non-causal (acausal) system. A non-causal system is also called a non-realizable system.

• E.g. y(n)=x(n2 ) , y(n)=x(-n) , y(n) = x(n/3), y(n)=x(n)+x(n+1)

18 January 2011 2

• A system that depends solely on future input values is an anticausal system.

• Eg: y(n) = x(n+1), prediction of current value from only future values in the corrupted CD.

• Fact: All anti-causal /non causal systems are memory systems but opposite is not true.

• To check always take negative, 0, positive values and specially -1 < value < 1

• Observations: Negative index, index scaling, and power of index represent non-causality.

18 January 2011 3

Classification: Causal/non-causal/anti-causal

Classification: Stable/unstable • The system is said to be stable if any bounded(amplitude)

input signal results in bounded output signal– bounded signals u(n) , e-an where a>0

• The system is said to be unstable if the system gives unbounded output signal in response to bounded input signal– Unbounded signals r(n) , n*u(n)

• E.g. Consider the DT system of the bank account

• This grows without bound, due to 1.01 multiplier. This system is unstable.

18 January 2011 4

]1[01.1][][ nynxny

Stable system

y[n] = (x[n])2

Suppose x[n] is limited to the range -10 < x[n] < 10?

Signal Operations

• Time Shifting – Delaying(n=n-k)– Advancing(n=n+k)

• Time Reversal: negate the index or time(n=-n). • Time Scaling

– In Discrete Time it can also term as Rate Changing– Sampling rate can be changed to up or down– Up sampling/ Down sampling

• Amplitude Scaling: each sample of the signal would be scaled by scaling• Addition/Subtraction: corresponding samples from both signals would be

added, subtracted• Multiplication: corresponding samples from both signals would be

multiplied

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6

t

x(t)

0

10

t

x(-t)

0

Example

10

So, x(-t) represents the time reversal (or inverse) of x(t).

The graph of x(-t) can be formed by rotating the graph of x(t) 180 about the y-axis(mirror image about y axis).

To time-reverse a signal, replace every t with –t.

Signal Operations: Time Reversing (inversion)

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Signal Operations: Time Reversing (inversion)

8

Example: Given x(t) below, sketch x(-t).

t

x(t)

1 20

10

3-1-2-3

t

x(-t)

1 20 3-1-2-3

Signal Operations: Time Reversing (inversion)

Signal Operations: Time Reversing

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Signal Operations: Time Shifting DT

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In general, a negative shift is a shift to the right (delaying). Similarly, a positive shift is a shift to the left (advancing).

Replacing every n in a waveform with n– N shifts the waveform N samples to the right.

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Signal Operations: Time Shifting CT

Signal Operations: Time Shifting CT

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t

x(t)

1 20

10

t

x2(t) = x(t + 1)

1-1 0

t

x1(t) = x(t - 1)

1 20 3

3

32

-1

-1

Example: Given x(t) below, sketch

x1(t) = x(t – 1) and x2(t) = x(t + 1).

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Time scaling

Time scaling is the compression or expansion of a signal.

Compressed signal

(t) = x(2t) is a compressed version of x(t) as shown on the right.

In general, (t) = x(at) represents a compressed signal if a > 1.

Expanded signal

Similarly, (t) = x(at) represents an expanded signal if a < 1.

t

x(t)

T20

10

t

(t) = x(t/2)

0

t

(t) = x(2t)

Original signal

0

T1

Expanded signal(a = 0.5)

Compressed signal(a = 2)

2T22T1

2

T1

2

T2

Example

10

10

To scale any function by a, replace each t by at in the function.

Signal Operations: Time Scaling

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Signal Operations: Time Scaling

15

Example: Given x(t) below, sketch x1(t) = x(2t) and x2(t) = x(0.5t) = x(t/2).

t

x(t)

1 20

10

t

x2(t) = x(t/2)

1-1 0

t

x1(t) = x(2t)

1 20 3

3

32

-1

-1

Signal Operations: Time Scaling

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Example: If x(t) = 10sin(4 t - ), sketch x(t), x1(t) = x(2t), and x2(t) = x(t/2).

Effect of time scaling on frequency: _________________________________

Effect of time scaling on amplitude: _________________________________

t

x(t)

0.5 1.00

t

x2(t) = x(t/2)

0.5-0.5 0

t

x1(t) = x(2t)

0.5 1.00 1.5

1.5

1.51.0

-0.5

-0.5

Signal Operations: Time Scaling

Signal Operations: Time Scaling

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TSh, TR, TS

• Order of operations– Time shifting(TSh)

– Time reversal(TR)

– Time scaling(TS)

– Amplitude scaling

• TSh and TR are not commutative.

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Amplitude Scaling

• It changes the amplitude of the signal by a scaling factor.

• Some amplifiers not only amplify signals but also add (or remove) a constant, or dc, value.

1/14/2011 19http://ptolemy.eecs.berkeley.edu/eecs20/berkeley/body.html

20

We can use various combinations of the three operations just covered: time shifting, time scaling, and time reversal. The operations can often be applied in different orders, but care must be taken.

Example: To form x(at - b) from x(t) we could use two approaches:

1) Time-shift then time-scale

A. Time-shift x(t) by b to obtain x(t - b). I.e., replace every t by t - b.

B. Time-scale x(t - b) by a (i.e., replace t by at) to form x(at - b)

2) Time-scale then time-shift

A. Time-scale x(t) by a to obtain x(at).

B. Time-shift x(at) by b/a (i.e., replace t with t – b/a) to yield

x(a[t – b/a]) = x(at – b)

Standard order is

(1) Time Shifting

(2) Time reversal

(3) Time scaling

Combined operations

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Example: Given x(t) below, sketch x1(t) = x(2t - 1) and x2(t) = x(t/2 + 1).

t

x(t)

1 20

10

t

x2(t) = x(t/2 + 1)

1-1 0

t

x1(t) = x(2t - 1)

1 20 3

3

32

-1

-1

Combined operations

Signal Addition

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Signal Addition: Saturation of color

• Color tree

• Moving along a radius of a circle changes the saturation(vividness) of a color(signal addition: white color is added.)

• Moving up the tree increases the lightness of a color

• Moving around a circle of given radius changes the hue of a color(different frequencies)

• These three coordinates can be described in terms of three numbers

• Photoshop: uses H, S and B

lightn

ess

hue

saturation

Acknowledgement

• Various graphics used here has been taken from public resources instead of redrawing it. Thanks to those who have created it.

• Thanks to:

– Prof. John G. Proakis

– Prof. Dimitris G. Manolakis