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Digital signal processing
Prepared By Md Ibrahim Khalil
• Signals are variables that carry information.
• It is described as a function of one or more
independent variables.
• Basically it is a physical quantity . It varies with some
independent or dependent variables.
• Signals can be One-dimensional or multi- dimensional.
Introduction to Signals.
Introduction to signals
A function of one or more variables that convey
information on the nature of a physical
phenomenon. Examples: v(t),i(t),x(t),heartbeat,
blood pressure, temperature, vibration.
1.A set of information or data.
2.Function of one or more independent variables.
3. Contains information about the behavior or
nature of some phenomenon
What is signal ?
 One-dimensional signals: function depends on a
single variable, e.g., speech signal.
 Multi-dimensional signals: function depends on two
or more variables, e.g., image
 Complex signal : A complex signal that has different
frequency and different amplitude .EX.
 𝑌=5𝑠𝑖𝑛1000𝜔𝑡+ 1/2 𝑐𝑜𝑠5000𝜔𝑡
Deterministic and random signals
Natural signal and Artificial Signal
Continuous-time and discrete-time
signals
Periodic and non-periodic signals
Casual and Non-casual signals
Even and odd signals
Energy and power signals
Classification of signals
 Behavior of these signals is predictable w.r.t time
 There is no uncertainty with respect to its value at any time.
These signals can be expressed mathematically.
 For example x(t) = sin(3t) is deterministic signal.
Deterministic signals
 Behavior of these signals is random i.e. not predictable w.r.t
time.
 There is an uncertainty with respect to its value at any time.
 These signals can’t be expressed mathematically.
 For example : Thermal Noise generated is non deterministic
signal. Random Signals:
Random signals
Natural signal are produced by naturally .
Example : Heart beat ,earthquake signal .
Artificial signal does not produce by naturally.
Natural and Artificial signals
Continuous Signals: Signals that are defined for all time instants.
Discrete signals: Signals that are defined only for certain time
instants: T, 2T, 3T, ….etc. (T) is the sampling time of the original
continuous signal.
Continuous V/S Discrete Signals
Even signals xe(t) and odd signals xo(t) are defined as
xe (t) = xe (−t) and xo (t) = −xo (−t).
Any signal is a sum of unique odd and even signals. Using
x(t) = xe(t)+xo(t) and x(−t) = xe(t) − xo(t), yields
xe(t) =0.5(x(t)+x(−t)) and xo(t) =0.5(x(t) − x(−t)).
Even & Odd Signals
 Even:
x(−t) = x(t)
x[−n] = x[n]
 Odd:
x(−t) = −x(t)
x[−n] = −x[n]
 Any signal x(t) can be expressed as
x(t) = xe(t) + xo(t)
x(−t) = xe(t) − xo(t)
where xe(t) = 1/2(x(t) + x(−t))
xo(t) = 1/2(x(t) − x(−t))
Even &Odd Signals
 A Continuous time signal is said to be periodic if it repeats at a
regular intervals.
 Non periodic signal do not repeat at regular time.
Periodic and Non periodic signal
 A signal is said to be an energy signal if its normalized
energy is nonzero and finite.
 For an energy signal: 0<E<infinite.
 A signal is said to be a power signal if it normalized power is
nonzero and finite.
 For a power signal: 0<p<-infinite.
Energy and Power signal
Systems process input signals to produce output signals .
• A system is combination of elements that manipulates one or
more signals to accomplish a function and produces some
output.
What is System ?
The Continuous Time & Discrete Time
systems are classified as follows:
 1.Static & Dynamic system
 2.Time invariant & Time variant system
 3.Linear & Non-linear system
 4.Causal & Non-causal system
 5.Stable & Unstable system
 6.Invertible & inverse system
Classification & properties of System
 Definition: The continuous time system is said to be
static if its output depends upon the present input
only.
 A static system is memory less system •
 It has no storage devices
 its output signal depends on present values of the
input signal.
Static system
Definition :The continuous time system is said to be
static if its output does not depend upon the present
input only.
 A dynamic system possesses memory
 It has the storage devices
 A system is said to possess memory if its output
signal depends on past values and future values of
the input signal.
 Dynamic system consist of integration ,
diffrentiation, delay elements or accumulation term
in their equation.
Dynamic System
*A system is said to be time invariant if a time delay
or time advance of the input signal leads to a
identical time shift in the output signal.
*Otherwise the system is called time variant system.
Time Invariant and Time Variant
Systems
 A system is said to be linear if it satisfies the principle of
superposition
 For checking the linearity of the given system, firstly we
check the response due to linear combination of inputs
 Then we combine the two outputs linearly in the same
manner as the inputs are combined and again total response
is checked
 If response in step 2 and 3 are the same,the system is linear
othewise it is non linear.
 Important point: Linear system produces zero output if
the input is zero under relaxed condition.
Linear & Non Linear Systems
 Definition: The system is said to be causal if its output at
any time depends upon present & past inputs only.
Example: y[n]=x[n]+1/2x[n-1]
 Non-causal system : A system is said to be non-causal if
the present value of the output signal depends only on the
future values of the input signal.
Example: y[n]=x[n+1]+1/2x[n-1]
 Important point: Normally all causal system are physically
realizable. There is no system which can generate the
output for inputs which will be available in future. Such
systems are non-causal, and they are not physically
realizable.
Causal & Non-causal system
 A system is said to be bounded-input bounded-
output stable (BIBO stable) if every bounded input
results in a bounded output.
 Otherwise it is called unstable system.
Stable & Unstable Systems
 It includes the transformation of independent
variables.
 It is performed in both continuous and discrete
time signals.
 Operations that are performed are
OPERATIONS ON SIGNALS
 Let two signals x(t) and y(t) are given,
 Their addition will be, z(t) = x(t) + y(t)
 Their substraction will be, z(t) = x(t) – y(t)
1.ADDITION &SUBSTRACTION
If a constant ‘A’ is given with a signal x(t)
z(t) = A.x(t)
 If A>1,it is an amplified signal.
 If A<1,it is an attenuated signal.
2.MULTIPLICATION OF SIGNAL BY A
CONSTANT
If two signals x(t) and y(t) are given , than their
multiplication will be
z(t) = x(t).y(t)
3.MULTIPLICATION OF TWO SIGNALS
 Let a signal x(t),than the signal x(t-T) represented a
delayed version of x(t),which is delayed by T sec.
4.SHIFTING IN TIME
Shifting of a signal in time
 adding or subtracting the amount of the shift to the time
variable in the function.
 x(t) → x(t–t )
 t > 0 (𝑡0 is positive value), signal is shifted to the right
(delay).
 𝑡0 < 0 (𝑡0 is negative value), signal is shifted to the left
(advance).
 x(t–2)? x(t) is delayed by 2 seconds.
 x(t+2)? x(t) is advanced by 2 seconds.
Signal Operations: Time Shifting
Signals and System
Signals and System
Signals and System
Signals and System

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Signals and System

  • 1. Digital signal processing Prepared By Md Ibrahim Khalil
  • 2. • Signals are variables that carry information. • It is described as a function of one or more independent variables. • Basically it is a physical quantity . It varies with some independent or dependent variables. • Signals can be One-dimensional or multi- dimensional. Introduction to Signals. Introduction to signals
  • 3. A function of one or more variables that convey information on the nature of a physical phenomenon. Examples: v(t),i(t),x(t),heartbeat, blood pressure, temperature, vibration. 1.A set of information or data. 2.Function of one or more independent variables. 3. Contains information about the behavior or nature of some phenomenon What is signal ?
  • 4.  One-dimensional signals: function depends on a single variable, e.g., speech signal.  Multi-dimensional signals: function depends on two or more variables, e.g., image  Complex signal : A complex signal that has different frequency and different amplitude .EX.  𝑌=5𝑠𝑖𝑛1000𝜔𝑡+ 1/2 𝑐𝑜𝑠5000𝜔𝑡
  • 5. Deterministic and random signals Natural signal and Artificial Signal Continuous-time and discrete-time signals Periodic and non-periodic signals Casual and Non-casual signals Even and odd signals Energy and power signals Classification of signals
  • 6.  Behavior of these signals is predictable w.r.t time  There is no uncertainty with respect to its value at any time. These signals can be expressed mathematically.  For example x(t) = sin(3t) is deterministic signal. Deterministic signals
  • 7.  Behavior of these signals is random i.e. not predictable w.r.t time.  There is an uncertainty with respect to its value at any time.  These signals can’t be expressed mathematically.  For example : Thermal Noise generated is non deterministic signal. Random Signals: Random signals
  • 8. Natural signal are produced by naturally . Example : Heart beat ,earthquake signal . Artificial signal does not produce by naturally. Natural and Artificial signals
  • 9. Continuous Signals: Signals that are defined for all time instants. Discrete signals: Signals that are defined only for certain time instants: T, 2T, 3T, ….etc. (T) is the sampling time of the original continuous signal. Continuous V/S Discrete Signals
  • 10. Even signals xe(t) and odd signals xo(t) are defined as xe (t) = xe (−t) and xo (t) = −xo (−t). Any signal is a sum of unique odd and even signals. Using x(t) = xe(t)+xo(t) and x(−t) = xe(t) − xo(t), yields xe(t) =0.5(x(t)+x(−t)) and xo(t) =0.5(x(t) − x(−t)). Even & Odd Signals
  • 11.  Even: x(−t) = x(t) x[−n] = x[n]  Odd: x(−t) = −x(t) x[−n] = −x[n]  Any signal x(t) can be expressed as x(t) = xe(t) + xo(t) x(−t) = xe(t) − xo(t) where xe(t) = 1/2(x(t) + x(−t)) xo(t) = 1/2(x(t) − x(−t)) Even &Odd Signals
  • 12.  A Continuous time signal is said to be periodic if it repeats at a regular intervals.  Non periodic signal do not repeat at regular time. Periodic and Non periodic signal
  • 13.
  • 14.  A signal is said to be an energy signal if its normalized energy is nonzero and finite.  For an energy signal: 0<E<infinite.  A signal is said to be a power signal if it normalized power is nonzero and finite.  For a power signal: 0<p<-infinite. Energy and Power signal
  • 15.
  • 16. Systems process input signals to produce output signals . • A system is combination of elements that manipulates one or more signals to accomplish a function and produces some output. What is System ?
  • 17. The Continuous Time & Discrete Time systems are classified as follows:  1.Static & Dynamic system  2.Time invariant & Time variant system  3.Linear & Non-linear system  4.Causal & Non-causal system  5.Stable & Unstable system  6.Invertible & inverse system Classification & properties of System
  • 18.  Definition: The continuous time system is said to be static if its output depends upon the present input only.  A static system is memory less system •  It has no storage devices  its output signal depends on present values of the input signal. Static system
  • 19. Definition :The continuous time system is said to be static if its output does not depend upon the present input only.  A dynamic system possesses memory  It has the storage devices  A system is said to possess memory if its output signal depends on past values and future values of the input signal.  Dynamic system consist of integration , diffrentiation, delay elements or accumulation term in their equation. Dynamic System
  • 20.
  • 21.
  • 22. *A system is said to be time invariant if a time delay or time advance of the input signal leads to a identical time shift in the output signal. *Otherwise the system is called time variant system. Time Invariant and Time Variant Systems
  • 23.  A system is said to be linear if it satisfies the principle of superposition  For checking the linearity of the given system, firstly we check the response due to linear combination of inputs  Then we combine the two outputs linearly in the same manner as the inputs are combined and again total response is checked  If response in step 2 and 3 are the same,the system is linear othewise it is non linear.  Important point: Linear system produces zero output if the input is zero under relaxed condition. Linear & Non Linear Systems
  • 24.  Definition: The system is said to be causal if its output at any time depends upon present & past inputs only. Example: y[n]=x[n]+1/2x[n-1]  Non-causal system : A system is said to be non-causal if the present value of the output signal depends only on the future values of the input signal. Example: y[n]=x[n+1]+1/2x[n-1]  Important point: Normally all causal system are physically realizable. There is no system which can generate the output for inputs which will be available in future. Such systems are non-causal, and they are not physically realizable. Causal & Non-causal system
  • 25.  A system is said to be bounded-input bounded- output stable (BIBO stable) if every bounded input results in a bounded output.  Otherwise it is called unstable system. Stable & Unstable Systems
  • 26.
  • 27.  It includes the transformation of independent variables.  It is performed in both continuous and discrete time signals.  Operations that are performed are OPERATIONS ON SIGNALS
  • 28.  Let two signals x(t) and y(t) are given,  Their addition will be, z(t) = x(t) + y(t)  Their substraction will be, z(t) = x(t) – y(t) 1.ADDITION &SUBSTRACTION
  • 29. If a constant ‘A’ is given with a signal x(t) z(t) = A.x(t)  If A>1,it is an amplified signal.  If A<1,it is an attenuated signal. 2.MULTIPLICATION OF SIGNAL BY A CONSTANT
  • 30. If two signals x(t) and y(t) are given , than their multiplication will be z(t) = x(t).y(t) 3.MULTIPLICATION OF TWO SIGNALS
  • 31.  Let a signal x(t),than the signal x(t-T) represented a delayed version of x(t),which is delayed by T sec. 4.SHIFTING IN TIME
  • 32. Shifting of a signal in time  adding or subtracting the amount of the shift to the time variable in the function.  x(t) → x(t–t )  t > 0 (𝑡0 is positive value), signal is shifted to the right (delay).  𝑡0 < 0 (𝑡0 is negative value), signal is shifted to the left (advance).  x(t–2)? x(t) is delayed by 2 seconds.  x(t+2)? x(t) is advanced by 2 seconds. Signal Operations: Time Shifting