chapter 1 discrete signals and systems

All right reserved. Copyright © 2019. FKEE, UTHM 1

CHAPTER 1DISCRETE SIGNALS AND

SYSTEMS

By: Sharifah Saon

1


ReviewAn analog signal is one in

which the signal intensity (y-axis) varies in a smooth

fashion over time, no breaks or discontinuities in the

signal

A digital signal is one which the signal intensity (y-axis) maintains a constant level

for some period of time and then changes to another

level Figure 1: (a) Analog Signal, (b) Digital Signal


Review

A continous signal will contain a value for all real numbers along the

time axis (x-axis)

A discrete signalwill only have

values at equally spaced interval

along the time axis (x-axis)


Signal processing touches our daily lives in more ways than we realize. There are countless applications and devices that utilize signal processing to make our technologies work.

https://signalprocessingsociety.org/our-story/signal-processing-101

How Signal Processing Affects Our Daily Lives


Real World Applications


❑ Other practical applications involving signal analysis, where the desired information is conveyed in digital form and no D/C converter is required

❑ Example– Digital processor of radar signals,

– the information (position of the aircraft or speed) extracted from the radar signal may simply be printed on paper


Discrete Signal and System

Part 1Introduction to DSP

Part 2Discrete Signal

Part 3Discrete System

❑ Advantages of digital system

❑ Elements of a DSP system

❑ DSP applications

❑ Discrete signal representation

❑ Type of discrete signals

❑ Common discrete signals

❑ Discrete signal operations

❑ Linearity property

❑ Time-invariance property

❑ Linear and time-invariant (LTI) discrete system


❑ Easy to analyze – using computers

❑ Efficient way of storing - CD

❑ Easy to simulate – cheap and safe

❑ Easy to maintain – quick and low cost

❑ Compact in size - p are small in size

❑ Robust – stable in presence of noise

❑ System work better

Why Digital


DSP ASP

Reconfiguration the

system operation.

Flexible; changing the

program

Implies a redesign, testing

and verification of the

hardware

Accuracy Much better to control Difficult to control because

of tolerance

Signal processing

algorithms

Allows the implementation of

more sophisticated algorithm

Difficult to perform precise

mathematical operations

Cost Cheaper Expensive

Limitations in DSP

The speed of operation of ADC and digital signal processor, especially for extremely

wide bandwidths signal, required fast-sampling-rate ADC and fast digital signal

processors.

Advantage of DSP over ASP


❑ Analog signal processing

❑ Digital signal processing -method for processing the analog signal

Element of DSP System

Analog Signal Processor

Analog output signal

Analog input signal

D/A converter

Analog output signal

Digital input signal

Digital output signal

Analog input signal

Nyquist Filter

Sample Hold

Digital signal

processing

A/D converter

FS

LPF


❑ Nyquist Filter (Anti-Aliasing filter)– limits the highest analog , choose the suitable sampling rate and

prevent aliasing (Detail in Chapter 3) ❑ Sample Hold

– convert analog to discrete signal ❑ A/D converter

– convert the analog (discrete) signal to digital that appropriate as an input to the digital processor

❑ DS Processor– is a large programmable digital computer or small processor

programmed to perform the desired operations on the input signal ❑ D/A converter

– as an interface to provide the analog signal to the user ❑ FS LPF

– smooth the reconstructed signal


Discrete Signals

❑ Discrete signal is a sampled signal.

❑May arise naturally.

❑ Or as a sequence of sampling continuous signals (e.g. at a uniform sampling interval 𝑡𝑠).

❑ Discrete signal 𝑥 𝑛 is an ordered sequence of values corresponding to the integer index n.

❑ 𝑥 𝑛 is plotted as lines against the index n.

st


0 5 10 15 20 25 30 35 40

Continuous-time signal

Discrete-time signal

1

- 1

1

- 1


❑ The origin 𝑡 = 0 corresponds to 𝑛 = 0.

❑ A marker () indicates the origin 𝑛 = 0.

❑ E.g. 𝑥 𝑛 = {1, 2, 4⇓

, 8, 5} , 𝑥 𝑛 = {3, 4⇓

, 5,⋯}

❑ Ellipses (⋯) denote infinite extent on either side.

Discrete Signal Representation


❑ Left-handed and right-handed signals

❑ Causal, non-causal and anti-causal signals

❑ Periodic signals

❑ Even and odd symmetrical signals

❑ Energy and power signals

Type of Discrete Signals


Type of signal

❑ 𝑥 𝑛 is called:

– right-sided if it is zero for 𝑛 < 𝑁.

– causal if it is zero for 𝑛 < 0.

– left-sided if it is zero for 𝑛 > 𝑁.

– anti-causal if it is zero for 𝑛 > 0.


Periodic signals

17All right reserved. Copyright © 2011. Sharifah Saon

A periodic discrete signal repeats every N samples is described

by:

x[n] = x[n ± kN], k = 0, 1, 2, 3, ...

❑ x[n]={1⇓

, 3, 6, 4, 1, 3, 6, 4,…} x[n]= x[n+k (4)],

❑ where k is number of cycle and N is number of sample in one

period

k = 0 k = 1

n 0,4,8,… 1,5,9,… 2,6,10,… 3,7,11,…

x[n] 1 3 6 4


❑ If a signal 𝑥 𝑛 is identical to its mirror image 𝑥 −𝑛 , it is called an even symmetric signal.

❑ If 𝑥 𝑛 differs from its mirror image 𝑥 −𝑛only in sign, it is called an odd symmetric signal or antisymmetric signal

Symmetrical Signal


❑ It extends over symmetric limits of −𝑁 < 𝑛 < 𝑁.

❑ For odd symmetric signal, 𝑥𝑜 0 = 0 & the sum of 𝑥𝑜 𝑛 over symmetric limits (– β, β) equals zero:

[ ] 0o

k

x k

=−

=

][][ nxnx ee −= ][][ nxnx oo −−=


❑ Even symmetry & odd symmetry are mutually exclusive.

❑ Any signal 𝑥 𝑛 may be expressed as the sum of 𝑥𝑒 𝑛 & 𝑥𝑜 𝑛 .

𝒙 𝒏 = 𝒙𝒆 𝒏 + 𝒙𝒐 𝒏 (a)

❑ To find 𝑥𝑒 𝑛 & 𝑥𝑜 𝑛 from 𝑥 𝑛 , 𝑥 𝑛 is folded & invoked symmetry to get:

𝒙 −𝒏 = 𝒙𝒆 −𝒏 + 𝒙𝒐 −𝒏 = 𝒙𝒆 𝒏 − 𝒙𝒐 𝒏 (b)

Even and Odd Parts of Discrete Signals


❑ Adding equation (a) & (b):

2𝒙𝒆 𝒏 = 𝒙 𝒏 + 𝒙 −𝒏

𝒙𝒆 𝒏 = 𝟎. 𝟓𝒙 𝒏 + 𝟎. 𝟓𝒙 −𝒏

❑ Subtracting equation (a) & (b):2𝒙𝒐 𝒏 = 𝒙 𝒏 − 𝒙 −𝒏𝒙𝒐 𝒏 = 𝟎. 𝟓𝒙 𝒏 − 𝟎. 𝟓𝒙 −𝒏

❑ If 𝒙 𝒏 has even symmetry, 𝒙𝒐 𝒏 will equal to zero.

❑ If 𝒙 𝒏 has odd symmetry, 𝒙𝒆 𝒏 will equal to zero.


Given

Calculate its odd and even parts.

Example 1.1

}6,4,2,4{][ −−=

nx


Solution

n −2 −1 0 1 2

x[n] 4 −2 4 −6

x[−n] −6 4 −2 4

0.5x[n] 2 −1 2 −3

0.5x[−n] −3 2 −1 2

xe[n] 2 −4 4 −4 2

xo[n] 2 2 0 −2 −2


Energy and Power Signals

❑ Signals with finite energy are called energy signals.

❑ Signals with finite power are called power signals.

❑ All periodic signals are power signals


❑ For non-periodic signals, the signal energy E

is a useful measure. It is defined as:

❑ For periodic signals, energy is .

Measurement of Discrete Signals

2[ ]

n

E x n

=−

=


❑Measures for periodic signals are based on averages since their signal energy is .

❑ Average value 𝑥𝑎𝑣 = average sum per period.

❑ Signal power, P = average energy per period.

−

=

=1

0

][1 N

n

av nxN

x

12

0

1[ ]

N

n

P x nN

−

=

=


Example 1.2

Calculate the energy in the signal

𝑥 𝑛 = 3 0.5 𝑛, 𝑛 ≥ 0.

Determine type of this discrete signal, whether the signal is an energy or power discrete signal?


❑ This describes a one-sided decaying exponential. Its signal energy is

❑ Note that

Solution

JnxEn

n

n

n

n

1225.01

9)25.0(9)5.0(3][

0

2

0

2 =−

====

=

=

−=

−=

= 1

1

0n

n


1 12 2

0 0

1 1[ ] [ ]

N

n n

P x n x nN

− −

= =

= = =

This signal has finite energy and infinite power. Therefore, it

can be classified as energy discrete signal.

To classify the signal, we need to calculate power of the

signal.


Exercise 1.1


❑ Unit impulse

❑ Unit step

❑ Unit ramp

❑ Rectangular

❑ Triangular

Common Discrete Signals


❑ unit impulse

❑ unit step

❑ unit ramp

Common Discrete Signals 0, 0

[ ]1, 0

nn

n

=

=

0, 0[ ]

1, 0

nu n

n

=

0, 0[ ] [ ]

, 0

nr n nu n

n n

= =


❑ The product of x[n] with impulse 𝛿[𝑛 – 𝑘] results in:

𝑥 𝑛 𝛿 𝑛 – 𝑘 = 𝑥[𝑘]𝛿[𝑛 – 𝑘]

❑ This is an impulse with strength 𝑥[𝑘]. The product property leads to:

❑ This is the shifting property. The impulse extracts the value 𝑥[𝑘] from 𝑥[𝑛] at impulse location 𝑛 = 𝑘.

Properties of the Discrete Impulse

−=

=−n

kxknnx ][][][


❑ Any 𝑥[𝑛] may be expressed as a sum of shifted impulses 𝛿[𝑛 – 𝑘] whose strengths 𝑥[𝑘] correspond to the signal values at 𝑛 = 𝑘. Thus,

𝑥[𝑛] =

𝑘=−∞

∞

]𝑥[𝑛]𝛿[𝑛 − 𝑘

❑ 𝑢[𝑛] & 𝑟[𝑛] may be expressed as a train of shifted impulses:

𝑢[𝑛] =

𝑘=0

∞

]𝛿[𝑛 − 𝑘

𝑟[𝑛] =

𝑘=0

∞

]𝑘𝛿[𝑛 − 𝑘

Signal Representation by Impulses


❑ u[n] may also be expressed as the cumulative sum of δ[n] & r[n] may be described as the cumulative sum of u[n]:

𝑢[𝑛] =

𝑘=−∞

𝑛

]𝛿[𝑘

𝑟[𝑛] =

𝑘=−∞

𝑛

]𝑢[𝑘


Discrete Pulse Signals

1,

2 0,

n Nnrect

N elsewhere

=

1 ,

0,

nn n N

tri NN

elsewhere

−

=


❑ 𝑟𝑒𝑐𝑡(𝑛/2𝑁) has 2𝑁 + 1 samples over

– 𝑁 < 𝑛 < 𝑁.

❑ 𝑡𝑟𝑖(𝑛/𝑁) has 2𝑁 + 1 samples over –𝑁 < 𝑛 < 𝑁, with the end samples 𝑥[𝑁] & 𝑥[–𝑁] being zero


Mathematically describe the signals below in at least two different ways

(a) (b) (c)

Example 1.3


Solution


2 r [n – 1]

2 r [n – 4]

6 u [n – 7]


Exercise 1.2

Synthesize the following pulse in term of or function:

x[n]

6

4

2

-2 -1 0 1 2 3 4 5 6

y[n]

n

6

4

2

-6 -5 -4 -3 -2 -1 0 1 2


3) [ ] 6

3

2) [ ] 5

2 6

n na x n tri tri

N

n nb y n rect rect

N

− = =

+ = =

Solution


Discrete Signals Operations❑ Common operations: addition & multiplication.

❑ Time shifting

❑ Folding / inversion

❑ Time scaling (decimation and interpolation)

❑ Addition

❑ Multiplication


Time Shifting

❑ 𝑦 𝑛 = 𝑥 𝑛 − 𝛽 for 𝛽 > 0 means a delayed version of 𝑥 𝑛 (shifted to the right by 𝛽)

❑ 𝑦 𝑛 = 𝑥 𝑛 + 𝛽 for 𝛽 > 0 means an advanced version of 𝑥 𝑛 (shifted to the left by 𝛽)


Folding

❑ The signal 𝑦 𝑛 = 𝑥 −𝑛 is a folded version of

𝑥 𝑛 (mirror image of 𝑥 𝑛 about 𝑛 = 0).

❑ 𝑥 −𝑛 − 𝛽 can be generated from 𝑥 𝑛 by:

1. Shift right β units: 𝑥 𝑛 ⇒ 𝑥 𝑛 − 𝛽 . • Then fold: 𝑥 𝑛 − 𝛽 ⇒ 𝑥 −𝑛 − 𝛽 .

2. Fold: 𝑥 𝑛 ⇒ 𝑥 − 𝑛 . • Then shift left β units: 𝑥 −𝑛 ⇒ 𝑥 − 𝑛 + 𝛽 = 𝑥 −𝑛 − 𝛽 .


Example 1.4

Given . Find and sketch:

1. y [n] = x [n – 3]

2. f [n] = x [n + 2]

3. g [n] = x [– n]

4. h [n] = x [– n + 1]

5. s [n] = x [– n – 2 ]

}7,6,5,4,3,2{

=nx


Solution

( )

1. 3 {0, 2, 3, 4, 5, 6,7}

2. 2 {2, 3, 4, 5, 6, 7}

3. {7, 6, 5, 4, 3, 2}

4. 1 {7, 6, 5, 4, 3, 2}(fold , then delay by 1)

5. 2 {7, 6, 5, 4, 3, 2} fold , then advance by 2

y n x n

f n x n

g n x n

h n x n x n

s n x n x n

= − =

= + =

= − =

= − + =

= − − =


Exercise 1.3


DECIMATION is a process of reducing the signal length by discarding signal samples –

downsampling

INTERPOLATION is a process of increasing the signal length by inserting signal samples -

upsampling

Decimation and Interpolation


❑ 𝑥 𝑛 is 𝑥 𝑡 sampled at 𝑡𝑠.

❑ Signal 𝑦[𝑛] = 𝑥[2𝑛] is the compressed signal 𝑥(2𝑡)sampled at 𝑡𝑠.

❑ It contains 𝑥[0], 𝑥[2], 𝑥[4],⋯ (alternate samples of 𝑥[𝑛])

❑ 𝑦[𝑛] can be obtained directly from 𝑥(𝑡) (not its compressed version) if we sample it at 2𝑡𝑠.

❑ It means a twofold reduction in the sampling rate. Decimation by a factor of 𝑁 is sampling 𝑥(𝑡) at intervals 𝑁𝑡𝑠 & implies N-fold reduction in the sampling rate

Decimation


❑ 𝑥[𝑛] is 𝑥(𝑡) sampled at intervals 𝑡𝑠.

❑ 𝑦[𝑛] = 𝑥[𝑛/2] is 𝑥(𝑡) sampled at 𝑡𝑠/2 & has twice the length of 𝑥[𝑛] with one new sample between adjacent samples of 𝑥[𝑛].

❑ If expression for 𝑥[𝑛] were known, it would be no problem to determine new sample values.

❑ Otherwise, the best to do is interpolatebetween samples.

Interpolation


❑ May choose each new sample value as:

– zero (zero interpolation)

– a constant equal to the previous sample value (step interpolation)

– average of adjacent sample values (linear interpolation)


❑ Zero interpolation is known as up-sampling& plays an important role in practical.

❑ Interpolation by factor N is equivalent to sampling 𝑥(𝑡) at intervals 𝑡𝑠/𝑁.

❑ It implies N-fold increase in both the sampling rate & the signal length


❑ Consider two sets of operations below:𝒙[𝒏]→ decimate by 𝟐 → 𝒙[𝟐𝒏]

→ interpolate by 𝟐 → 𝒙[𝒏]

𝒙[𝒏]→ interpolate by 𝟐 → 𝒙[𝒏/𝟐]→ decimate by 𝟐 → 𝒙[𝒏]

❑ Both operations appear to recover 𝑥[𝑛] suggesting interpolation & decimation are inverse operations.

❑ In fact, only the second sequence of operations (interpolation followed by decimation) recovers 𝑥[𝑛]exactly!


Given

Calculate two sequence of operation, by using step interpolation (𝑛/2).

Example 1.5

}8,4,6,2,1{][

=nx


Solution

Note: If both interpolation and decimation are required, it is better to

interpolate first.


❑ If a signal x[n] is interpolated by N and then decimated by N, we recover the original signal x[n] will be recovered.

❑ If a signal x[n] is first decimated by N and then interpolated by N, x[n] may not recovered.

❑ If both interpolation and decimation are required, it is better to interpolate first.


i. Let . Generate 𝑥[2𝑛] and various interpolated versions of 𝑥[𝑛/3]

ii. Let .

Calculate 𝑔[𝑛] = 𝑥[2𝑛 – 1] and the step interpolated signal ℎ 𝑛 = 𝑥[0.5𝑛 − 1].

iii. Let .

Determine 𝑦[𝑛] = 𝑥[2𝑛/3].

Exercise 1.4

}1,5,2,1{][ −=

nx

}6,5,4,3{][

=nx

}6,5,4,3{][

=nx


❑ Fractional delays (typically half-sample) can be implemented using interpolation & decimation.

❑ Fractional delay of 𝒙[𝒏] requires interpolation, shift & decimation (in that order).

❑ 𝑥[𝑛]→ → means

interpolate 𝑥[𝑛] by 𝑁, delay by 𝑀, then decimate by 𝑁.

Fractional Delay

−

N

Mnx

−

N

MNnx


Given

Calculate the signal 𝑦[𝑛] = 𝑥[𝑛 – 0.5] using linear interpolation.

Example 1.6

[ ] {2,4, 6, 8}.x n

=


❑ First interpolate by 2, delay by 1 and then decimate by 2.– Linear interpolation:

(last sample interpolated to zero).

– Delay:

– Decimation:

Solution

[ ] [ / 2] {2, 3, 4, 5, 6, 7, 8, 4,0}g n x n

= =

1[ ] [ 1]

2

{2, 3, 4, 5, 6, 7, 8, 4}

nh n g n x

− = − =

=

2 1[ ] [ 0.5] [2 ] {3, 5, 7, 4}

2

ny n x n x h n

− = − = = =


❑ Given

Analyse

a. 𝑠 𝑛 = 𝑥 𝑛 − 1.5

b. 𝑧 𝑛 = 𝑦 𝑛/2 where 𝑦 𝑛 = 𝑥 −𝑛 + 2

Consider linear interpolation if applicable.

Exercise 1.5

}3,8,6,4{][

=nx


❑ Collection of hardware components, or software routines, that operates on a discrete-time signal sequence.

❑ Systems are modeled as transformations or operators that map input signals to outputs

– Continuous Time : 𝑦 𝑡 = 𝑇 𝑥 𝑡

– Discrete Time : 𝑦 𝑛 = 𝑇 𝑥 𝑛

❑ 𝑇 . denotes the operator or transformation that models the effect of the system on the input signal

Discrete Systems


❑ Classes of systems are defined by which of the following characteristics they possess

– Memory

– Linearity

– Time-Invariance

– Causality

– Stability

– Linear Time-Invariant (LTI)

❑ These characteristics place constraints on the corresponding system operators or transformations

System Classes


Linearity property of a discrete system

A discrete system is linear if

For any signals , and constants and .

Otherwise, the system is called nonlinear.

T a1x1 n[ ]+a2x2 n[ ]{ } =a1T x1 n[ ]{ }+a2T x2 n[ ]{ }

x1 n[ ] x2 n[ ] a1 a2


Time-invariance property of a discrete system

A time-invariant discrete system is a system whose input-output behaviordoes not change with time. Time-invariant system is also called as relaxedsystem and shift-invariant system.

The discrete system is time-invariant if

implies that

For every input signal 𝑥[𝑛] and time shift, 𝑘.

Otherwise, the system is called time-variant.

x n[ ] T¾ ®¾ y n[ ] x n- k[ ] T¾ ®¾ y n- k[ ]


Linear Time-Invariant (LTI) Discrete System

Input

signal

Output

signalLTI Discrete system

Linear time-invariant discrete systems (LTI systems) are a class of systems used in DSP that are both linear and time-invariant.

Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to

those inputs. Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI

systems easy to represent and understand graphically.


Linear Time-Invariant (LTI) System

LTI SystemINPUT OUTPUT

.T


Exercise: Test Sem 2 20182019

chapter 1 discrete signals and systems

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