EC 2314 Digital Signal Processing

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EC 2314 Digital Signal Processing

• By
• Dr. K. Udhayakumar

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Signal

• A signal is a pattern of variation that carry
  information.
• Signals are represented mathematically as a
  function of one or more independent variable
• A picture is brightness as a function of two
  spatial variables, x and y.
• In this course signals involving a single
  independent variable, generally refer to as a
  time, t are considered. Although it may not
  represent time in specific application
• A signal is a real-valued or scalar-valued
  function of an independent variable t.

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Signal Types
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Signal Types

• Analog signals continuous in time and amplitude
• Example voltage, current, temperature,
• Digital signals discrete both in time and
  amplitude
• Example attendance of this class, digitizes
  analog signals,
• Discrete-time signals discrete in time,
  continuous in amplitude
• Example hourly change of temperature
• Theory of digital signals would be too
  complicated
• Requires inclusion of nonlinearities into theory
• Theory is based on discrete-time
  continuous-amplitude signals
• Most convenient to develop theory
• Good enough approximation to practice with some
  care
• In practice we mostly process digital signals on
  processors
• Need to take into account finite precision effects

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Signal Types

• Continuous time
• Continuous amplitude
• Continuous time
• Discrete amplitude
• Discrete time
• Continuous amplitude
• Discrete time
• Discrete amplitude

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Example of signals

• Electrical signals like voltages, current and EM
  field in circuit
• Acoustic signals like audio or speech signals
  (analog or digital)
• Video signals like intensity variation in an
  image
• Biological signal like sequence of bases in gene
• Noise which will be treated as unwanted signal

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Signal classification

• Continuous-time and Discrete-time
• Energy and Power
• Real and Complex
• Periodic and Non-periodic
• Analog and Digital
• Even and Odd
• Deterministic and Random

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A continuous-time signal

• Continuous-time signal x(t), the independent
  variable, t is Continuous-time. The signal itself
  needs not to be continuous.

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Continuous Time (CT) Signals

• Most signals in the real world are continuous
  time, as the scale is infinitesimally fine.
• E.g. voltage, velocity,
• Denote by x(t), where the time interval may be
  bounded (finite) or infinite

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A piecewise continuous-time signal

• A piecewise continuous-time signal

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Discrete Time (DT) Signals

• Some real world and many digital signals are
  discrete time, as they are sampled
• E.g. pixels, daily stock price (anything that a
  digital computer processes)
• Denote by xn, where n is an integer value that
  varies discretely
• Sampled continuous signal
• xn x(nk)

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A discrete-time signal

• A discrete signal is defined only at
  discrete instances. Thus, the independent
  variable has discrete values only.

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Sampling

• A discrete signal can be derived from a
  continuous-time signal by sampling it at a
  uniform rate.
• If denotes the sampling period and denotes
  an integer that may assume positive and negative
  values,
• Sampling a continuous-time signal x(t) at time
  yields a sample of value
• For convenience, a discrete-time signal is
  represented by a sequence of numbers
• We write
• Such a sequence of numbers is referred to as a
  time series.

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Periodic Signals

• An important class of signals is the class of
  periodic signals. A periodic signal is a
  continuous time signal x(t), that has the
  property
• where Tgt0, for all t.
• Examples
• cos(t2p) cos(t)
• sin(t2p) sin(t)
• Are both periodic with period 2p

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Odd and Even Signals

• An even signal is identical to its time reversed
  signal, i.e. it can be reflected in the origin
  and is equal to the original
• Examples
• x(t) cos(t)
• x(t) c
• An odd signal is identical to its negated, time
  reversed signal, i.e. it is equal to the negative
  reflected signal
• Examples
• x(t) sin(t)
• x(t) t
• This is important because any signal can be
  expressed as the sum of an odd signal and an even
  signal.

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Exponential and Sinusoidal Signals

• Exponential and sinusoidal signals are
  characteristic of real-world signals and also
  from a basis (a building block) for other
  signals.
• A generic complex exponential signal is of the
  form
• where C and a are, in general, complex numbers.
  Lets investigate some special cases of this
  signal
• Real exponential signals

Exponential growth
Exponential decay
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Periodic Complex Exponential Sinusoidal Signals

• Consider when a is purely imaginary
• By Eulers relationship, this can be expressed
  as
• This is a periodic signals because
• when T2p/w0
• A closely related signal is the sinusoidal
  signal
• We can always use

cos(1)
T0 2p/w0 p
T0 is the fundamental time period w0 is the
fundamental frequency
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Exponential Sinusoidal Signal Properties

• Periodic signals, in particular complex periodic
  and sinusoidal signals, have infinite total
  energy but finite average power.
• Consider energy over one period
• Therefore
• Average power
• Useful to consider harmonic signals
• Terminology is consistent with its use in music,
  where each frequency is an integer multiple of a
  fundamental frequency

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General Complex Exponential Signals

• So far, considered the real and periodic complex
  exponential
• Now consider when C can be complex. Let us
  express C is polar form and a in rectangular
  form
• So
• Using Eulers relation
• These are damped sinusoids

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Discrete Unit Impulse and Step Signals

• The discrete unit impulse signal is defined
• Useful as a basis for analyzing other signals
• The discrete unit step signal is defined
• Note that the unit impulse is the first
  difference (derivative) of the step signal
• Similarly, the unit step is the running sum
  (integral) of the unit impulse.

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Continuous Unit Impulse and Step Signals

• The continuous unit impulse signal is defined
• Note that it is discontinuous at t0
• The arrow is used to denote area, rather than
  actual value
• Again, useful for an infinite basis
• The continuous unit step signal is defined

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A piecewise discrete-time signal

• A piecewise discrete-time signal

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Energy and Power Signals

• X(t) is a continuous power signal if
• Xn is a discrete power signal if
• X(t) is a continuous energy signal if
• Xn is a discrete energy signal if

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Power and Energy in a Physical System

• The instantaneous power
• The total energy
• The average power

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Energy and Power over Infinite Time

• For many signals, were interested in examining
  the power and energy over an infinite time
  interval (-8, 8). These quantities are therefore
  defined by
• If the sums or integrals do not converge, the
  energy of such a signal is infinite
• Two important (sub)classes of signals
• Finite total energy (and therefore zero average
  power)
• Finite average power (and therefore infinite
  total energy)
• Signal analysis over infinite time, all depends
  on the tails (limiting behaviour)

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Power and Energy

• By definition, the total energy over the time
  interval in a continuous-time
  signal is
• denote the magnitude of the (possibly
  complex) number
• The time average power
• By definition, the total energy over the time
  interval in a discrete-time
  signal is
• The time average power

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Power and Energy

• Example 1
• The signal is given below is energy or
  power signal.
• Explain.
• This signal is energy signal

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Power and Energy

• Example 2
• The signal is given below is energy
  or power signal.
• Explain.
• This signal is energy signal

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Real and Complex

• A value of a complex signal is a
  complex number
• The complex conjugate, of the
  signal is
• Magnitude or absolute value
• Phase or angle

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Periodic and Non-periodic

• A signal or is a periodic
  signal if
• Here, and are fundamental period,
  which is the smallest positive values when
• Example

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Analog and Digital

• Digital signal is discrete-time signal whose
  values belong to a defined set of real numbers
• Binary signal is digital signal whose values are
  1 or 0
• Analog signal is a non-digital signal

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Even and Odd

• Even Signals
• The continuous-time signal
  /discrete-time signal is an even
  signal if it satisfies the condition
• Even signals are symmetric about the vertical
  axis
• Odd Signals
• The signal is said to be an odd signal if it
  satisfies the condition
• Odd signals are anti-symmetric (asymmetric) about
  the time origin

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Even and Odd signalsFacts

• Product of 2 even or 2 odd signals is an even
  signal
• Product of an even and an odd signal is an odd
  signal
• Any signal (continuous and discrete) can be
  expressed as sum of an even and an odd signal

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Complex-Valued Signal Symmetry

• For a complex-valued signal
• is said to be conjugate symmetric if it
  satisfies the condition
• where
• is the real part and is the imaginary
  part
• is the square root of -1

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Deterministic and Random signal

• A signal is deterministic whose future values can
  be predicted accurately.
• Example
• A signal is random whose future values can NOT be
  predicted with complete accuracy
• Random signals whose future values can be
  statistically determined based on the past values
  are correlated signals.
• Random signals whose future values can NOT be
  statistically determined from past values are
  uncorrelated signals and are more random than
  correlated signals.

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Deterministic and Random signal(contd)

• Two ways to describe the randomness of the signal
  are
• Entropy
• This is the natural meaning and mostly used in
  system performance measurement.
• Correlation
• This is useful in signal processing by
  directly using correlation functions.

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Basic sequences and sequence operations

• Delaying (Shifting) a sequence
• Unit sample (impulse) sequence
• Unit step sequence
• Exponential sequences

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Discrete-Time Systems

• A Discrete-Time System is a mathematical
  operation that maps a given input sequence xn
  into an output sequence yn
• Example
• Moving (Running) Average
• Maximum
• Ideal Delay System

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Memoryless System

• A system is memoryless if the output yn at
  every value of n depends only on the input xn
  at the same value of n
• Example
• Square
• Sign
• counter example
• Ideal Delay System

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Linear Systems

• Linear System A system is linear if and only if
• Example Ideal Delay System

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Time-Invariant Systems

• Time-Invariant (shift-invariant) Systems
• A time shift at the input causes corresponding
  time-shift at output
• Example Square
• Counter Example Compressor System

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Causal System

• A system is causal iff its output is a function
  of only the current and previous samples
• Examples Backward Difference
• Counter Example Forward Difference

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Stable System

• Stability (in the sense of bounded-input
  bounded-output BIBO). A system is stable iff
  every bounded input produces a bounded output
• Example Square
• Counter Example Log

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LTI System Example
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Linear Time-Invariant Systems

• Special importance for their mathematical
  tractability
• Most signal processing applications involve LTI
  systems
• LTI system can be completely characterized by
  their impulse response

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• Ex)

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Properties of LTI Systems

• Convolution is commutative
• Convolution is distributive

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Properties of LTI Systems

• Cascade connection of LTI systems

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Stable and Causal LTI Systems

• An LTI system is (BIBO) stable iff Impulse
  response is absolute summable
• Lets write the output of the system as
• Then the output is bounded by
• The output is bounded if the absolute sum is
  finite
• An LTI system is causal iff

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Stability Condition A linear time-invariant
system is stable If and only if
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Causality Condition
Neither necessary nor sufficient condition for
all systems, But necessary and sufficient for
LTI system
But xn-k for kgt0 shows The future values of
xn. So yn depends only on the Future values
of xn.
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Linear Constant-Coefficient Difference Equations

• An important class of LTI systems of the form
• The output is not uniquely specified for a given
  input
• The initial conditions are required
• Linearity, time invariance, and causality depend
  on the initial conditions
• If initial conditions are assumed to be zero
  system is linear, time invariant, and causal
• Example
• Moving Average

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• Linear Constant-Coefficient Difference Equations
• Ex)

Xn is the difference of yn
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• Frequency-Domain Representation

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• Ex)

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• Ex)

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Eigenfunctions of LTI Systems

• Complex exponentials are eigenfunctions of LTI
  systems
• Lets see what happens if we feed xn into an
  LTI system
• The eigenvalue is called the frequency response
  of the system
• is a complex function of
  frequency

Eigenfunction
Eigenvalue
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Discrete-Time Fourier Transform

• Many sequences can be expressed as a weighted sum
  of complex exponentials as
• Where the weighting is determined as
• is the Fourier spectrum of the
  sequence xn
• The phase wraps at 2? hence is not uniquely
  specified
• The frequency response of a LTI system is the
  DTFT of the impulse response

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Absolute and Square Summability

• For a given sequence if the infinite sum
  convergence, the DTFT exist
• All stable systems are absolute summable and have
  finite and continues frequency response

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Absolute and Square Summability

• Absolute summability is sufficient condition for
  DTFT
• Some sequences may not be absolute summable but
  only square summable
• Such sequences can be represented by fourier
  transform if
• In other words, the error
  may not approach zero at each
  value of as but the total
  energy in the error does.

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Example Ideal Lowpass Filter

• The periodic DTFT of the ideal lowpass filter is
• The inverse can be written as
• Not causal, Not absolute summable but it has a
  DTFT, The DTFT converges in the mean-squared
  sense
• Role of Gibbs phenomenon

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Ex)
The impulse response is not causal, Not
absolutely summable, but squarely summable, Since
sequence values approach zero as n-gt
infinity, But only as 1/n
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Symmetric Sequence and Functions
Conjugate-symmetric Conjugate-antisymmetric
Sequence

Function

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Exploiting Superposition and Time-Invariance

• Are there sets of basic signaxkn, such that
• We can represent any signal as a linear
  combination (e.g, weighted sum) of these building
  blocks? (Hint Recall Fourier Series.)
• The response of an LTI system to these basic
  signals is easy to compute and provides
  significant insight.
• For LTI Systems (CT or DT) there are two natural
  choices for these building blocks
• Later we will learn that there are many families
  of such functions sinusoids, exponentials, and
  even data-dependent functions. The latter are
  extremely useful in compression and pattern
  recognition applications.

• CT Systems(impulse)

• DT Systems(unit pulse)

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Representation of DT Signals Using Unit Pulses
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Response of a DT LTI Systems Convolution

• Define the unit pulse response, hn, as the
  response of a DT LTI system to a unit pulse
  function, ?n.
• Using the principle of time-invariance
• Using the principle of linearity
• Comments
• Recall that linearity implies the weighted sum of
  input signals will produce a similar weighted sum
  of output signals.
• Each unit pulse function, ?n-k, produces a
  corresponding time-delayed version of the system
  impulse response function (hn-k).
• The summation is referred to as the convolution
  sum.
• The symbol is used to denote the convolution
  operation.

convolution operator
convolution sum
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LTI Systems and Impulse Response

• The output of any DT LTI is a convolution of the
  input signal with the unit pulse response
• Any DT LTI system is completely characterized by
  its unit pulse response.
• Convolution has a simple graphical interpretation

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Visualizing Convolution

• There are four basic steps to the calculation
• The operation has a simple graphical
  interpretation

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Calculating Successive Values

• We can calculate each output point by shifting
  the unit pulse response one sample at a time

• yn 0 for n lt ???
• y-1
• y0
• y1
• yn 0 for n gt ???
• Can we generalize this result?

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Graphical Convolution
2
1
-1
-1
1
-1
k -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
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Graphical Convolution (Cont.)
2
1
-1
-1
1
-1
k -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
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Graphical Convolution (Cont.)

• Observations
• yn 0 for n gt 4
• If we define the duration of hn as the
  difference in time from the first nonzero sample
  to the last nonzero sample, the duration of hn,
  Lh, is4 samples.
• Similarly, Lx 3.
• The duration of yn is Ly Lx Lh 1. This
  is a good sanity check.
• The fact that the output has a duration longer
  than the input indicates that convolution often
  acts like a low pass filter and smoothes the
  signal.

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Examples of DT Convolution

• Example unit-pulse

• Example delayed unit-pulse

• Example unit step

• Example integration

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Properties of Convolution

• Commutative

• Implications

• Distributive

• Associative

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Useful Properties of (DT) LTI Systems

• Causality

• Stability

Bounded Input ? Bounded Output
Sufficient Condition
Necessary Condition
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Convolution Representation..Example

• Consider the DT system described by
• Its impulse response can be found to be

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Representing Signals in Terms ofShifted and
Scaled Impulses

• Let xn be an arbitrary input signal to a DT LTI
  system
• Suppose that for
• This signal can be represented as

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Exploiting Time-Invariance and Linearity
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The Convolution Sum

• This particular summation is called the
  convolution sum
• Equation is called
  the convolution representation of the system
• Remark a DT LTI system is completely described
  by its impulse response hn

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Block Diagram Representation of DT LTI Systems

• Since the impulse response hn provides the
  complete description of a DT LTI system, we write

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The Convolution Sum for Noncausal Signals

• Suppose that we have two signals xn and vn
  that are not zero for negative times (noncausal
  signals)
• Then, their convolution is expressed by the
  two-sided series

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Example Convolution of Two Rectangular Pulses

• Suppose that both xn and vn are equal to the
  rectangular pulse pn (causal signal) depicted
  below

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The Folded Pulse

• The signal is equal to the pulse pi
  folded about the vertical axis

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Sliding over
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Sliding over - Contd
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Plot of
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Properties of the Convolution Sum

• Associativity
• Commutativity
• Distributivity w.r.t. addition

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Properties of the Convolution Sum - Contd

• Shift property define
• Convolution with the unit impulse
• Convolution with the shifted unit impulse

then
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Changing the Sampling rate using discrete-time
processing

• downsampling sampling rate compressor

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Frequency domain of downsampling

• Since this is a re-sampling process. Remember
  that, from continuous-time sampling of
  xnxc(nT), we have
• Similarly, for the down-sampled signal
  xdmxc(mT), (where T MT), we have

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Frequency domain of downsampling

• We are interested in the relation between X(ejw)
  and Xd(ejw). Lets represent r as r i kM,
  where 0 ? i ? M?1, (i.e., r ? i (mod M)). Then

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Frequency domain of downsampling

• Therefore, the downsampling can be treated as a
  re-sampling process. It s frequency domain
  relationship is similar to that of the D/C
  converter as
• This is equivalent to compositing M copies of the
  of X(ejw), frequency scaled by M and shifted by
  inter multiples of 2?.
• The aliasing can be avoided by ensuring that
  X(ejw) is bandlimited as

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Example of downsampling in the Frequency domain
(without aliasing)
Sampling with a sufficiently large rate which
avoids aliasing
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Example of downsampling in the Frequency domain
(without aliasing)
Downsampling by 2 (M2)
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Downsampling with prefiltering to avoid aliasing
(decimation)

• From the above, the DTFT of the down-sampled
  signal is the superposition of M shifted/scaled
  versions of the DTFT of the original signal.
• To avoid aliasing, we need wNlt?/M, where wN is
  the highest frequency of the discrete-time signal
  xn.
• Hence, downsampling is usually accompanied with a
  pre-low-pass filtering, and a low-pass filter
  followed by down-sampling is usually called a
  decimator, and termed the process as decimation.

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Up-sampling

• Upsampling sampling rate expander.
• or equivalently,
• In frequency domain

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Example of up-sampling
Upsampling in the frequency domain
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Up-sampling with post low-pass filtering

• Similar to the case of D/C converter, upsampoling
  is usually companied with a post low-pass filter
  with cutoff frequency ?/L and gain L, to
  reconstruct the sequence.
• A low-pass filter followed by up-sampling is
  called an interpolator, and the whole process is
  called interpolation.

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Example of up-sampling followed by low-pass
filtering
Applying low-pass filtering
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Interpolation

• Similar to the ideal D/C converter,
• If we choose an ideal lowpass filter with cutoff
  frequency ?/L and gain L, its impulse response is
  
• Hence

Its an interpolation of the discrete sequence xk
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Sample and hold
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Example of sample and hold
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Quantizer (Quantization)

• The real-valued signal has to be stored as a code
  for digital processing. This step is called
  quantization.
• The quantizer is a nonlinear system.
• Typically, we apply twos complement code for
  representation.

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Quantizer (Quantization)
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Quantizer (Quantization)

• In general, if we have a (B1)-bit binary twos
  complement fraction of the form
• then its value is
• The step size of the quantizer is
• where Xm is the full scale level of the A/D
  converter.
• The numerical relationship beween the code words
  and the quantizer samples is

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Example of quantization
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Analysis of quantization errors

• Quantization error
• In general, for a (B1)-bit quantizer with step
  size ?, the quantization error satisfies that
• when
• If xn is outside this range, then the
  quantization error is larger in magnitude than
  ?/2, and such samples are saided to be clipped.

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Analysis of quantization errors

• Analyzing the quantization by introducing an
  error source and linearizing the system
• The model is equivalent to quantizer if we know
  en.

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Assumptions about en

• en is a sample sequence of a stationary random
  process.
• en is uncorrelated with the sequence xn.
• The random variables of the error process en
  are uncorrelated i.e., the error is a
  white-noise process.
• The probability distribution of the error process
  is uniform over the range of quantization error
  (i.e., without being clipped).
• The assumptions would not be justified. However,
  when the signal is a complicated signal (such as
  speech or music), the assumptions are more
  realistic.
• Experiments have shown that, as the signal
  becomes more complicated, the measured
  correlation between the signal and the
  quantization error decreases, and the error also
  becomes uncorrelated.

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Example of quantization error
original signal
3-bit quantization result
3-bit quantization error
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Example of quantization error
8-bit quantization error

• In a heuristic sense, the assumptions of the
  statistical model appear to be valid if the
  signal is sufficiently complex and the
  quantization steps are sufficiently small, so
  that the amplitude of the signal is likely to
  traverse many quantization steps from sample to
  sample.

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Quantization error analysis

• en is a white noise sequence. The probability
  density function of en is

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Quantization error analysis

• The mean value of en is zero, and its variance
  is
• Since
• For a (B1)-bit quantizer with full-scale
  value Xm, the noise variance, or power, is

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Quantization error analysis

• A common measure of the amount of degradation of
  a signal by additive noise is the signal-to-noise
  ratio (SNR), defined as the ratio of signal
  variance (power) to noise variance. Expressed in
  decibels (dB), the SNR of a (B1)-bit quantizer
  is
• Hence, the SNR increases approximately 6dB for
  each bit added to the world length of the
  quantized samples.

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Quantization error analysis

• The equation can be further simplified for
  analysis. For example, if the signal amplitude
  has a Gaussian distribution, only 0.064 percent
  of the samples would have an amplitude greater
  than 4?x.
• Thus to avoid clipping the peaks of the signal
  (as is assumed in our statistical model), we
  might set the gain of filters and amplifiers
  preceding the A/D converter so that ?x Xm/4.
  Using this value of ?x gives
• For example, obtaining a SNR about 90-96 dB in
  high-quality music recording and playback
  requires 16-bit quantization.
• But it should be remembered that such performance
  is obtained only if the input signal is carefully
  matched to the full-scale of the A/D converter.

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Spectral Analysis

• Spectral analysis is concerned with the
  determination of the energy or power spectrum of
  a continuous-time signal
• It is assumed that is sufficiently
  bandlimited so that its spectral characteristics
  are reasonably estimated from those of its of its
  discrete-time equivalent gn

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Spectral Analysis

• To ensure bandlimited nature is
  initially filtered using an analogue
  anti-aliasing filter the output of which is
  sampled to provide gn
• Assumptions
• (1) Effect of aliasing can be ignored
• (2) A/D conversion noise can be neglected

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Spectral Analysis

• Three typical areas of spectral analysis are
• 1) Spectral analysis of stationary sinusoidal
  signals
• 2) Spectral analysis of of nonstationary signals
• 3) Spectral analysis of random signals

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Spectral Analysis of Sinusoidal Signals

• Assumption - Parameters characterising sinusoidal
  signals, such as amplitude, frequency, and phase,
  do not change with time
• For such a signal gn, the Fourier analysis can
  be carried out by computing the DTFT

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Spectral Analysis of Sinusoidal Signals

• Initially the infinite-length sequence gn is
  windowed by a length-N window wn to yield
• DTFT of then is assumed
  to provide a reasonable estimate of
• is evaluated at a set of R (
  ) discrete angular frequencies using
  an R-point FFT

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Spectral Analysis of Sinusoidal Signals

• Note that
• The normalised discrete-time angular frequency
  corresponding to DFT bin k is
• while the equivalent continuous-time angular
  frequency is

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Sampling and Aliasing..Overview

• Periodic sampling, the process of representing a
  continuous signal with a sequence of discrete
  data values, pervades the field of digital signal
  processing.
• In practice, sampling is performed by applying a
  continuous signal to an analog-to-digital (A/D)
  converter whose output is a series of digital
  values.

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Cont..

• With regard to sampling, the primary concern is
  how fast must the given continuous signal be
  sampled in order to preserve its information
  content.

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ALIASING

• There is a frequency-domain ambiguity associated
  with the discrete-time signal samples that is
  absent in the continuous signal world.

eg. Suppose you are given the following sequence
of values,
x(0) 0 x(1) 0.866 x(2) 0.866 x(3) 0
x(4) -0.866 x(5) -0.866 x(6) 0
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Oversampling
If the original waveform does not vary much over
the duration of p(t), then we will also obtain a
good construction. Oversampling, i.e., using a
sampling rate that is much greater than the
Nyquist rate, can ensure this.
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Spectral Analysis of Sinusoidal Signals

• Consider
• expressed as
• Its DTFT is given by

130
Spectral Analysis of Sinusoidal Signals

• is a periodic function of w with
  a period 2p containing two impulses in each
  period
• In the range , there is
  an impulse at
• of complex amplitude
  and an impulse at of complex
  amplitude
• To analyze gn using DFT, we employ a
  finite-length version of the sequence given by

131
Spectral Analysis of Sinusoidal Signals

• Example - Determine the 32-point DFT of a
  length-32 sequence gn obtained by sampling at a
  rate of 64 Hz a sinusoidal signal of
  frequency 10 Hz
• Since Hz the DFT bins will be
  located in Hz at ( k/NT)2k, k0,1,2,..,63
• One of these points is at given signal frequency
  of 10Hz

132
Spectral Analysis of Sinusoidal Signals

• DFT magnitude plot

133
Spectral Analysis of Sinusoidal Signals

• Example - Determine the 32-point DFT of a
  length-32 sequence gn obtained by sampling at a
  rate of 64 Hz a sinusoid of frequency 11 Hz
• Since
• the impulse at f 11 Hz of the DTFT appear
  between the DFT bin locations k 5 and k 6
• the impulse at f -11 Hz appears between the DFT
  bin locations k 26 and k 27

134
Spectral Analysis of Sinusoidal Signals

• DFT magnitude plot
• Note Spectrum contains frequency components at
  all bins, with two strong components at k 5 and
  k 6, and two strong components at k 26 and k
  27

135
Spectral Analysis of Sinusoidal Signals

• The phenomenon of the spread of energy from a
  single frequency to many DFT frequency locations
  is called leakage
• Problem gets more complicated if the signal
  contains more than one sinusoid

136
Spectral Analysis of Sinusoidal Signals

• Example
• -
• From plot it is difficult to determine if there
  is one or more sinusoids in xn and the exact
  locations of the sinusoids

137
Spectral Analysis of Sinusoidal Signals

• An increase in resolution and accuracy of the
  peak locations is obtained by increasing DFT
  length to R 128 with peaks occurring at k
  27 and k 45

138
Spectral Analysis of Sinusoidal Signals

• Reduced resolution occurs when the difference
  between the two frequencies becomes less than 0.4
• As the difference between the two frequencies
  gets smaller, the main lobes of the individual
  DTFTs get closer and eventually overlap

139
Spectral Analysis of Nonstationary Signals

• An example of a time-varying signal is the chirp
  signal and
  shown below for
• The instantaneous frequency of xn is

140
Spectral Analysis of Nonstationary Signals

• Other examples of such nonstationary signals are
  speech, radar and sonar signals
• DFT of the complete signal will provide
  misleading results
• A practical approach would be to segment the
  signal into a set of subsequences of short length
  with each subsequence centered at uniform
  intervals of time and compute DFTs of each
  subsequence

141
Spectral Analysis of Nonstationary Signals

• The frequency-domain description of the long
  sequence is then given by a set of short-length
  DFTs, i.e. a time-dependent DFT
• To represent a nonstationary xn in terms of a
  set of short-length subsequences, xn is
  multiplied by a window wn that is stationary
  with respect to time and move xn through the
  window

142
Spectral Analysis of Nonstationary Signals

• Four segments of the chirp signal as seen through
  a stationary length-200 rectangular window

143
Short-Time Fourier Transform

• Short-time Fourier transform (STFT), also known
  as time-dependent Fourier transform of a signal
  xn is defined by
• where wn is a suitably chosen window sequence
• If wn 1, definition of STFT reduces to that
  of DTFT of xn

144
Short-Time Fourier Transform

• is a function of 2
  variables integer time index n and continuous
  frequency w
• is a periodic function
  of w with a period 2p
• Display of is the
  spectrogram
• Display of spectrogram requires normally three
  dimensions

145
Short-Time Fourier Transform

• Often, STFT magnitude is plotted in two
  dimensions with the magnitude represented by the
  intensity of the plot
• Plot of STFT magnitude of chirp sequence
• with
  for a length of 20,000 samples
  computed using a Hamming window of length 200
  shown next

146
Short-Time Fourier Transform

• STFT for a given value of n is essentially the
  DFT of a segment of an almost sinusoidal sequence

147
Short-Time Fourier Transform

• Shape of the DFT of such a sequence is similar to
  that shown below
• Large nonzero-valued DFT samples around the
  frequency of the sinusoid
• Smaller nonzero-valued DFT samples at other
  frequency points

148
STFT on Speech

• An example of a narrowband spectrogram of a
  segment of speech signal

149
STFT on Speech

• The wideband spectrogram of the speech signal is
  shown below
• The frequency and time resolution tradeoff
  between the two spectrograms can be seen

150
DSP applications

• Speech, audio
• Noise reduction (Dolby), compression (MP3),
• Radar
• filtering, movement detection,
• Image processing
• Compression, pattern recognition, segmentation,
• Biomedical
• Monitoring, analysis, tele-medicine,