Signal

Any physical quantity that varies with time, space, or

ECE 411 any other independent variable or variables.
A function of one or more independent variables.
Signals, Spectra, and
Signal Processing Examples: d
a (t )
t2
Introduction u (t ) B sin( 2 ft )
s ( x, y ) 3 x 5 xy 2 y 2

2 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Block Diagram of an Analog

System Signal Processing System
An aggregate or assemblage of things so combined by Electrical Amplified
signal signal
nature or man as to form an aggregate or complex
whole.
Processor
A physical device (or software) that performs an
operation on a signal.
It is characterized by the type of operation that it
performs on the signal. Sound
Electrical
Electrical
transducer Amplifier
transducer
Bigger sound

3 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 4 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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 Block Diagram of an Digital
Digital Signal Processor Signal Processing System
A digital signal processor Electrical
signal
Amplified
signal
(DSP) is an integrated
circuit designed for high-
Processor
speed data
manipulations, and is
used in audio,
Electrical
communications, image Sound
transducer Electrical
transducer
manipulation, and other Bigger sound

data acquisition and

data control Analog Analog-to-Digital Digital Signal Digital-to-Analog Analog
LPF Converter Processor Converter LPF
applications.
Algorithm (Program)

5 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 6 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Advantages of Digital Signal Processing Advantages of Analog Signal Processing

Stability Low cost and simplicity in some applications
Low sensitivity to component tolerances Infinite resolution in amplitude
Low sensitivity to temperature changes Low signal levels
Low sensitivity to aging effects Infinite effective sampling rate
Higher noise immunity Wide bandwidth (up to GHz)
Programmability / Versatility
Repeatability

7 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 8 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Signal Representation
All electrical signals can be visualized using two basic
methods, the time domain and the frequency domain.
Time Domain
The time domain is the form of visualization that most people
are familiar with. Time domain
The most common time domain instrument is the
oscilloscope, which has graduations of volts on the y-axis and
graduations of time on the x-axis.
Frequency Domain
Instead of showing the variation of a signal with respect to
time, it shows the variation of the signal with respect to
frequency.
The most common instrument for displaying the frequency
domain is the spectrum analyzer. Frequency domain
9 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 10 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Real-World Signals Classification of Signals

Real-world signals, i.e., sound or speech, are typically Multi-channel Signal
more complicated than a simple sine wave. Multidimensional Signal
They consist of many frequencies of differing Continuous-time Signal
amplitudes that combine into a composite signal. Discrete-time Signal
Continuous-valued (or continuous-amplitude) Signals
Discrete-valued (or discrete-amplitude) Signals
Deterministic Signals
Random Signals

11 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 12 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Multi-channel Signal Multidimensional Signal
Signal generated by multiple sources which is Signal which is a function of two or more independent
represented in vector form. variables.

Example: S ( x, y, z ) x 2 y 2 xz 2 3 y 2
s1 (t )
S (t ) s2 (t )
Signals can be represented as a multi-channel,
multidimensional signal.
s3 (t ) I r ( x, y , t )
I ( x, y , t ) I g ( x , y , t )

I b ( x, y , t )
13 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 14 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Continuous-time Signals Discrete-time Signals

Defined for every value of time and they take on values Are defined only at certain specific values of time.
in the continuous interval (a, b), where a can be - and Are defined only at discrete points along t.
b can be +.
Also known as analog signals and it is described by
functions of a continuous variable. Example:

a tn
Example: x(t ) sin t , t y (t n ) 2e , n 0, 1, 2,
a t
y (t ) 2e , t

15 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 16 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Continuous-valued (continuous-amplitude) Signals Discrete-valued (discrete-amplitude) Signals
Signal takes on all possible values on a finite or an Signal takes on values from a finite set of possible
infinite range. values.

17 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 18 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Deterministic Signals Random Signals

Signals that can be uniquely described by an explicit Signals that cannot be described by a mathematical
mathematical expression, a table of data, or a well- formula.
defined rule.
Examples: noise, speech

19 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 20 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Continuous-time Sinusoidal Signals
Tp

xa (t ) A cos(t ), - t t

where: 1 cycle

A amplitude Period (Tp) Time to complete one cycle, sec

frequency in radians/sec Frequency (F) number of cycles per second, Hz
phase in radians x cycles 1
t sec
Tp F but Hz
cycle t sec sec
xa (t ) A cos(2Ft ), - t 1

1

sec
Tp
F cycle cycle
sec

21 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 22 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Properties of Continuous-time Sinusoids Discrete-time Sinusoidal Signals

For every fixed value of the frequency F, xa(t) is
x(n) A cos(n ), - n
1.
periodic.
xa (t Tp ) xa (t ) where Tp 1
F where:
2. Continuous-time sinusoidal signals with distinct n integer variable
frequencies are themselves distinct. A amplitude
frequency in radians/sample
3. Increasing the frequency F results in an increase in phase in radians
the rate of oscillation of the signal.
x(n) A cos(2fn ), - n

23 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 24 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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 Properties of Discrete-time Sinusoids
N
1. A discrete-time sinusoid is periodic only if its
n
frequency f is a rational number.
2. Discrete-time sinusoids whose frequencies are
1 cycle
separated by an integer multiple of 2 are identical.
Period (N) Number of samples to complete one cycle 3. The highest rate of oscillation in a discrete-time
n samples sinusoid is attained when = (or = ).
N
cycle

1 1
Since = Frequency , then f
Period N

25 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 26 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Block Diagram of Analog-to-Digital Conversion Sampling

The conversion of a continuous-time signal into a
discrete-time continuous-valued signal by taking
samples of the continuous-time signal.

Sampling Hold
gate circuit
Analog Sample and Digital
Quantizer Coder
input Hold output t n

CT signal DT signal

Sampling pulse

27 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 28 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Periodic or Uniform Sampling Tp

t Continuous-time
Type of sampling wherein the CT signal is sampled at signal, x(t)
equal intervals defined by the sampling period Ts. 1 cycle
t=0

Ts
x(n) = xa(nTs), -<n<
t Sampling pulses

n
th

th
nd

th

th
rd

th
0

1s
t
th

Ts = 1/Fs

pu
pu
pu

pu

pu

pu

pu
pu
where:

pu

ls e
ls e
lse

ls e

ls e

ls e

ls e
lse
t=0

lse
Fs sampling frequency
N

note that t = nTs = n/Fs 3 4 5

n Sampled signal
1 2 6 7 nth sample
(DT signal), x(n)
1 cycle new cycle
n=0
29 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 30 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Sampling Theorem
Given an analog signal Consider any analog signal represented as a sum of
xa (t ) A cos(2Ft ) sinusoids of different amplitudes, frequencies, and
phases:
xa (nTs ) A cos(2FnTs ) therefore, N
xa (t ) Ai cos(2Fi t i )
2Fn
xa (nTs ) A cos i 1

sF where: N number of frequency components

comparing this to the basic discrete-time sinusoidal
signal Sampling theorem states that: Fs 2Fmax
x(n) A cos( 2fn )
where: Fmax highest frequency component
F Fs - sampling rate
f or Ts
Fs

31 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 32 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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 Effect of Undersampling and Aliasing

1 Consider a compound signal

therefore, Fmax Fs or max Fs

x(t ) x1 (t ) x2 (t )
2 2Ts Ts
1 1 1 where: x1 (t ) cos(2F1t )
since F , then T p 2Ts x2 (t ) cos(2F2t )
Tp T p 2Ts
F2 3F1
If we sample the signal based on F1, then Fs = 2F1
Fmax F 1
Also, f max max or max 2 f max
Fs 2 Fmax 2
x1 (n) cos 2 F1
2 F1

n cos(n)

x (n) cos2
2
F2
2 F1
n cos2 3 F1
2 F1

n cos(3n)

33 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 34 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

t x(t ) x1 (t ) x2 (t )
To avoid undersampling, an analog low-pass filter
(usually Butterworth) designed around the systems
t=0 desired Fmax is placed before the sampling gate.
n
x1 (n) cos(n)
This LPF filters out all unwanted frequencies above Fmax
x1(0) = 0 x1(1) = x1(2) = 2 from the input signal reducing* the effect of aliasing on
the sampled signal for a given Fs.
* Analog filters have a gradual roll-off. A steeper roll-off (higher
n
x2 (n) cos(3n) order) is more desirable, but affects the phase response of
x2(0) = 0 x2(1) = 3 x2(2) = 6 the filter.
This filter is known as an anti-aliasing filter.
x1(n) and x2(n) are identical!
x2(n) is an alias of x1(n).
35 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 36 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Quantization Quantization of Discrete-time Continuous-valued Signals

The conversion of a discrete-time continuous-valued Let

signal to a discrete-time discrete-valued signal. x q ( n ) Q [ x ( n )]
Quantization Error the difference between the where:
unquantized sample to the quantized signal. Also called xq(n) denotes the sequence of quantized samples
quantization noise. Q[x(n)] denotes the quantizer operation on the samples x(n)
Raw Quantized Absolute
Sample
value value Error
3
x(0) 2.3 2 0.3
Two possible operations:
2

1
x(1) 1.2 1 0.2 Truncation process of eliminating the undesired excess digits
x(2) 0.2 0 0.2
0 n by assigning each sample to the quantization level below it.
-1 x(3) -1.3 -2 0.7
-2 x(4) -2.4 -3 0.6 Rounding process of assigning each sample to the nearest
-3 x(5) -1.2 -2 0.8
quantization level by eliminating the undesired excess digits
x(6) 0.1 0 0.1
x(7) 1.1 1 0.1 (if over), or by adding extra digits (if under).

37 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 38 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Coding
Dynamic Range difference between the minimum (xmin) Conversion of discrete-time discrete-valued signal to a
and maximum (xmax) unquantized signal, x(n). binary sequence (e.g. 1011011).
Quantization level allowable values in the digital signal. b log2 L
where:
Quantization step or Resolution, distance between two
b number of coding bits
successive quantization levels. Also called step size.
L number of quantization levels
x x min
max Value Binary Equiv.
L 1 3 011
2 010
Quantization Error, eq(n) difference between the 1 001

unquantized sample to quantized signal. 0 000

-1 101
-2 110
eq (n) x(n) xq (n) -3 111

39 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 40 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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Digital-to-Analog Conversion
Low-pass

Digital-to-Analog
Serial-to-Parallel
Filter

Converter
Converter
A digital-to-analog converter (DAC) converts a digital 10011010 t
signal (binary value) to an analog equivalent. Serial data

The value would be held at the output of the DAC until Reconstructed Analog Signal
a new value arrives (defined by the sampling interval, Clock signal
Ts). The result is a stair-step representation of the
3 Reconstructed signal
digital signal (zero-order hold). 2
(interpolated)

Another analog LPF smoothes-out the square wave by 1

Original signal

filtering out the higher harmonics above the systems 0 t

defined Fmax. This filter is called a postfilter or a -1

-2
reconstruction filter. -3

Stair-step (zero-order hold) output from DAC

41 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 42 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

Issues in Digital Signal Processing

Sampling Processing Speed
The Nyquist-Shannon theorems minimum sample rate may A big issue especially for real-time applications.
not be adequate for some types of signals. n-length data words need system clock rates of at least n
To totally eliminate aliasing, ideal LPFs (brick wall, zero-phase times the sampling rate.
shift) are needed before the sampling gate. High sampling rates and high resolution data produce large
amounts of data that require very fast processors.
Quantization and Coding
Processing speed also depends on algorithm efficiency.
Finite-precision binary words limit the resolution of the signal.
The length of the data word (8, 16, 24bit) determines the
DAC and Analog Reconstruction
number of quantization levels (and thus resolution). DAC components must be carefully matched to produce
accurate results.
The limited resolution introduces quantization noise and
errors in the computational results. The performance of LPFs as reconstruction filters (tracking)
depend on the time constant of the filter, and can introduce
a phase delay on the reconstructed signal.
43 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 44 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

11
 1. Given a continuous-time signal

xa(t)=3cos100t

ECE 411 a) Determine the minimum sampling rate required to avoid

Signals, Spectra, and aliasing.
b) Suppose that the signal is sampled at the rate
Signal Processing Fs = 200 Hz. What is the discrete-time signal obtained after
sampling?
Problem Set (Introduction) c) Suppose that the signal is sampled at the rate of
Fs = 75 Hz. What is the discrete-time signal obtained after
sampling?
d) What is the frequency 0 F Fs/2 of a sinusoid that yields
samples identical to those obtained in part (c)?

46 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

2. Given an analog signal 3. Determine whether or not each of the following signals
is periodic. In case a signal is periodic, specify its
fundamental period.
xa(t) = 3cos 2000t + 5sin 6000t + 10cos 12000t
a) xa(t) = 3 cos (5t + /6)
b) x(n) = cos (0.01n)
a) What is the Nyquist rate for this signal?
c) x(n) = 3 cos (5n + /6)
b) What is the discrete-time signal obtained after sampling
using Fs = 5000 samples/sec? d) x(n) = cos (n/3) cos (n/8)
c) What is the analog signal ya(t) we can reconstruct from the
samples if we use ideal D/A converter ?

47 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 48 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

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4. The discrete-time signal
ASSIGNMENT 1:
x(n) = 6.35cos[(/10)n]
Answer the problems at the end of Chapter 1
(problems 1.1 up to 1.10) on the textbook (Digital
is quantized with resolution
Signal Processing, Proakis/Manolakis, 3e).
a) = 0.1
Use an A4-sized paper
b) = 0.02
Handwritten (including the problem)
Submission is on our next meeting
How many bits are required in the A/D converter in
each case? Determine the noise floor and signal-to-
noise (S/N) ratio in dB as well.

49 ECE 411 - Signals, Spectra, and Signal Processing: Introduction 50 ECE 411 - Signals, Spectra, and Signal Processing: Introduction

13