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UNIT - 4 ANALYSIS OF DISCRETE TIME SIGNALS
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Sampling Frequency Harry Nyquist, working at Bell Labs developed what has become known as the Nyquist Sampling Theorem: – In order to be ‘perfectly’ represented by its samples, a signal must be sampled at a sampling rate (also called sampling frequency) equal to at least twice its highest frequency component – Or: f s = 2f – Note that f s here is frequency of sampling, not the frequency of the sample How often do you sample? The sampling rate depends on the signal’s highest frequency (for baseband)
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Sampling Rate Examples Take Concert A: 440 Hz – What would be the minimum sampling rate needed to accurately capture this signal? – f s = 2 x 440 Hz = 880 Hz Take your telephone used for voice, mostly – Highest voice component is: 3000 Hz – Minimum sampling rate: f s = 2 x 3000 Hz = 6000 Hz – Real telephone digitization is done at 8000 Hz sampling rate (supporting a 4 kHz bandwidth). Why? Remember that Nyquist said “equal to at least twice…”
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Undersampling and Oversampling Undersampling – Sampling at an inadequate frequency rate – Aliased into new form - Aliasing – Loses information in the original signal Oversampling – Sampling at a rate higher than minimum rate – More values to digitize and process – Increases the amount of storage and transmission – COST $$
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Effects of Undersampling Original waveform Reconstructed waveform
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DISCRETE TIME FOURIER TRANSFROM (DTFT)
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FREQUENCY SPECTRUM
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Derivation of the Discrete-time Fourier Transform
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Recall DTFS pair where
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The limit of integration is over any interval of 2 in Periodic in with period 2 Thus,
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DTFT Pair
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Conditions for Convergence
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Examples
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IDTFT
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6) Complex Exponentials
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DTFT of Periodic Signals Recall the following DTFT pair: Represent periodic signal x[n] in terms of DTFS:
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Example: A discrete-time Sine Function
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Example: A discrete-time Periodic Impulse Train The DTFS coefficients for this signal are: ckck
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Properties of DTFT
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Convolution Property
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Multiplication Property
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PROPERTIES OF DTFT
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The z-Transform
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Content Introduction z-Transform Zeros and Poles Region of Convergence Important z-Transform Pairs Inverse z-Transform z-Transform Theorems and Properties System Function
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The z-Transform Introduction
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Why z-Transform? A generalization of Fourier transform Why generalize it? – FT does not converge on all sequence – Notation good for analysis – Bring the power of complex variable theory deal with the discrete-time signals and systems
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The z-Transform z-Transform
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Definition The z-transform of sequence x(n) is defined by Let z = e j . Fourier Transform
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z-Plane Re Im z = e j Fourier Transform is to evaluate z-transform on a unit circle.
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z-Plane Re Im X(z)X(z) Re Im z = e j
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Periodic Property of FT Re Im X(z)X(z) X(ej)X(ej) Can you say why Fourier Transform is a periodic function with period 2 ?
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The z-Transform Zeros and Poles
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Definition Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|< , is called the region of convergence. ROC is centered on origin and consists of a set of rings.
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Example: Region of Convergence Re Im ROC is an annual ring centered on the origin. r
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Stable Systems A stable system requires that its Fourier transform is uniformly convergent. Re Im 1 Fact: Fourier transform is to evaluate z-transform on a unit circle. A stable system requires the ROC of z-transform to include the unit circle.
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Example: A right sided Sequence 12345678910-2-3-4-5-6-7-8 n x(n)x(n)...
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Example: A right sided Sequence For convergence of X(z), we require that
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a aa Example: A right sided Sequence ROC for x(n)=a n u(n) Re Im 1 a aa Re Im 1 Which one is stable?
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Example: A left sided Sequence 12345678910-2-3-4-5-6-7-8 n x(n)x(n)...
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Example: A left sided Sequence For convergence of X(z), we require that
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a aa Example: A left sided Sequence ROC for x(n)= a n u( n 1) Re Im 1 a aa Re Im 1 Which one is stable?
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The z-Transform Region of Convergence
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Represent z-transform as a Rational Function where P(z) and Q(z) are polynomials in z. Zeros: The values of z’s such that X(z) = 0 Poles: The values of z’s such that X(z) =
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Example: A right sided Sequence Re Im a ROC is bounded by the pole and is the exterior of a circle.
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Example: A left sided Sequence Re Im a ROC is bounded by the pole and is the interior of a circle.
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Example: Sum of Two Right Sided Sequences Re Im 1/2 1/3 1/12 ROC is bounded by poles and is the exterior of a circle. ROC does not include any pole.
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Example: A Two Sided Sequence Re Im 1/2 1/3 1/12 ROC is bounded by poles and is a ring. ROC does not include any pole.
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Example: A Finite Sequence Re Im ROC: 0 < z < ROC does not include any pole. N-1 poles N-1 zeros Always Stable
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Properties of ROC A ring or disk in the z-plane centered at the origin. The Fourier Transform of x(n) is converge absolutely iff the ROC includes the unit circle. The ROC cannot include any poles Finite Duration Sequences: The ROC is the entire z-plane except possibly z=0 or z= . Right sided sequences: The ROC extends outward from the outermost finite pole in X(z) to z= . Left sided sequences: The ROC extends inward from the innermost nonzero pole in X(z) to z=0.
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More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Find the possible ROC’s
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More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 1: A right sided Sequence.
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More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 2: A left sided Sequence.
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More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 3: A two sided Sequence.
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More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 4: Another two sided Sequence.
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The z-Transform Important z-Transform Pairs
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Z-Transform Pairs Sequencez-TransformROC All z All z except 0 (if m>0) or (if m<0)
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Z-Transform Pairs Sequencez-TransformROC
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The z-Transform Inverse z-Transform
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The Inverse Z-Transform Formal inverse z-transform is based on a Cauchy integral Less formal ways sufficient most of the time – Inspection method – Partial fraction expansion – Power series expansion
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Inspection Method – Make use of known z-transform pairs such as Example: The inverse z-transform of
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Inverse Z-Transform by Partial Fraction Expansion Assume that a given z-transform can be expressed as Apply partial fractional expansion First term exist only if M>N – B r is obtained by long division Second term represents all first order poles Third term represents an order s pole – There will be a similar term for every high-order pole Each term can be inverse transformed by inspection
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Partial Fractional Expression Coefficients are given as
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Example: 2 nd Order Z-Transform – Order of nominator is smaller than denominator (in terms of z -1 ) No higher order pole
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Example Continued ROC extends to infinity – Indicates right sided sequence
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Example #2 Long division to obtain B o
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Example #2 Continued ROC extends to infinity – Indicates right-sides sequence
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Inverse Z-Transform by Power Series Expansion The z-transform is power series In expanded form Z-transforms of this form can generally be inversed easily Especially useful for finite-length series
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Example,
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Z-Transform Properties: Linearity Notation Linearity – Note that the ROC of combined sequence may be larger than either ROC – This would happen if some pole/zero cancellation occurs – Example: Both sequences are right-sided Both sequences have a pole z=a Both have a ROC defined as |z|>|a| In the combined sequence the pole at z=a cancels with a zero at z=a The combined ROC is the entire z plane except z=0
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Z-Transform Properties: Time Shifting Here n o is an integer – If positive the sequence is shifted right – If negative the sequence is shifted left The ROC can change the new term may – Add or remove poles at z=0 or z= Example
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Z-Transform Properties: Multiplication by Exponential ROC is scaled by |z o | All pole/zero locations are scaled If z o is a positive real number: z-plane shrinks or expands If z o is a complex number with unit magnitude it rotates Example: We know the z-transform pair Let’s find the z-transform of
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Z-Transform Properties: Differentiation Example: We want the inverse z-transform of Let’s differentiate to obtain rational expression Making use of z-transform properties and ROC Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing86
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Z-Transform Properties: Conjugation Example Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing87
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Z-Transform Properties: Time Reversal ROC is inverted Example: Time reversed version of Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing88
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Z-Transform Properties: Convolution Convolution in time domain is multiplication in z-domain Example:Let’s calculate the convolution of Multiplications of z-transforms is ROC: if |a| 1 if |a|>1 ROC is |z|>|a| Partial fractional expansion of Y(z) Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing89
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The z-Transform z-Transform Theorems and Properties
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Linearity Overlay of the above two ROC’s
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Shift
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Multiplication by an Exponential Sequence
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Differentiation of X(z)
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Conjugation
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Reversal
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Real and Imaginary Parts
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Initial Value Theorem
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Convolution of Sequences
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The z-Transform System Function
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Shift-Invariant System h(n)h(n) h(n)h(n) x(n)x(n) y(n)=x(n)*h(n) X(z)X(z)Y(z)=X(z)H(z) H(z)H(z)
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Shift-Invariant System H(z)H(z) H(z)H(z) X(z)X(z) Y(z)Y(z)
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N th -Order Difference Equation
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Representation in Factored Form Contributes poles at 0 and zeros at c r Contributes zeros at 0 and poles at d r
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Stable and Causal Systems Re Im Causal Systems : ROC extends outward from the outermost pole.
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Stable and Causal Systems Re Im Stable Systems : ROC includes the unit circle. 1
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Example Consider the causal system characterized by Re Im 1 a
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Determination of Frequency Response from pole-zero pattern A LTI system is completely characterized by its pole-zero pattern. Example: Re Im z1z1 p1p1 p2p2
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Determination of Frequency Response from pole-zero pattern A LTI system is completely characterized by its pole-zero pattern. Example: Re Im z1z1 p1p1 p2p2 |H(e j )|=? H(e j )=?
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Determination of Frequency Response from pole-zero pattern A LTI system is completely characterized by its pole-zero pattern. Example: Re Im z1z1 p1p1 p2p2 |H(e j )|=? H(e j )=? |H(e j )| = | 11 22 33 H(e j ) = 1 ( 2 + 3 )
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Example Re Im a dB
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