UNIT - 4 ANALYSIS OF DISCRETE TIME SIGNALS

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)

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…”

 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 $$

Effects of Undersampling Original waveform Reconstructed waveform

DISCRETE TIME FOURIER TRANSFROM (DTFT)

FREQUENCY SPECTRUM

Derivation of the Discrete-time Fourier Transform

Recall DTFS pair where

The limit of integration is over any interval of 2  in   Periodic in  with period 2  Thus,

DTFT Pair

Conditions for Convergence

Examples

IDTFT

6) Complex Exponentials

DTFT of Periodic Signals Recall the following DTFT pair: Represent periodic signal x[n] in terms of DTFS:

Example: A discrete-time Sine Function

Example: A discrete-time Periodic Impulse Train The DTFS coefficients for this signal are: ckck

Properties of DTFT

Convolution Property

Multiplication Property

PROPERTIES OF DTFT

The z-Transform

Content Introduction z-Transform Zeros and Poles Region of Convergence Important z-Transform Pairs Inverse z-Transform z-Transform Theorems and Properties System Function

The z-Transform Introduction

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

The z-Transform z-Transform

Definition The z-transform of sequence x(n) is defined by Let z = e  j . Fourier Transform

z-Plane Re Im z = e  j   Fourier Transform is to evaluate z-transform on a unit circle.

z-Plane Re Im X(z)X(z) Re Im z = e  j  

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  ?

The z-Transform Zeros and Poles

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.

Example: Region of Convergence Re Im ROC is an annual ring centered on the origin. r

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.

Example: A right sided Sequence n x(n)x(n)...

Example: A right sided Sequence For convergence of X(z), we require that

a aa Example: A right sided Sequence ROC for x(n)=a n u(n) Re Im 1 a aa Re Im 1 Which one is stable?

Example: A left sided Sequence n x(n)x(n)...

Example: A left sided Sequence For convergence of X(z), we require that

a aa Example: A left sided Sequence ROC for x(n)=  a n u(  n  1) Re Im 1 a aa Re Im 1 Which one is stable?

The z-Transform Region of Convergence

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) = 

Example: A right sided Sequence Re Im a ROC is bounded by the pole and is the exterior of a circle.

Example: A left sided Sequence Re Im a ROC is bounded by the pole and is the interior of a circle.

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.

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.

Example: A Finite Sequence Re Im ROC: 0 < z <  ROC does not include any pole. N-1 poles N-1 zeros Always Stable

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.

More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Find the possible ROC’s

More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 1: A right sided Sequence.

More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 2: A left sided Sequence.

More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 3: A two sided Sequence.

More on Rational z-Transform Re Im abc Consider the rational z-transform with the pole pattern: Case 4: Another two sided Sequence.

The z-Transform Important z-Transform Pairs

Z-Transform Pairs Sequencez-TransformROC All z All z except 0 (if m>0) or  (if m<0)

Z-Transform Pairs Sequencez-TransformROC

The z-Transform Inverse z-Transform

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

Inspection Method – Make use of known z-transform pairs such as Example: The inverse z-transform of

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

Partial Fractional Expression Coefficients are given as

Example: 2 nd Order Z-Transform – Order of nominator is smaller than denominator (in terms of z -1 ) No higher order pole

Example Continued ROC extends to infinity – Indicates right sided sequence

Example #2 Long division to obtain B o

Example #2 Continued ROC extends to infinity – Indicates right-sides sequence

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

Example,

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

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

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

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

Z-Transform Properties: Conjugation Example Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing87

Z-Transform Properties: Time Reversal ROC is inverted Example: Time reversed version of Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing88

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

The z-Transform z-Transform Theorems and Properties

Linearity Overlay of the above two ROC’s

Shift

Multiplication by an Exponential Sequence

Differentiation of X(z)

Conjugation

Reversal

Real and Imaginary Parts

Initial Value Theorem

Convolution of Sequences

The z-Transform System Function

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)

Shift-Invariant System H(z)H(z) H(z)H(z) X(z)X(z) Y(z)Y(z)

N th -Order Difference Equation

Representation in Factored Form Contributes poles at 0 and zeros at c r Contributes zeros at 0 and poles at d r

Stable and Causal Systems Re Im Causal Systems : ROC extends outward from the outermost pole.

Stable and Causal Systems Re Im Stable Systems : ROC includes the unit circle. 1

Example Consider the causal system characterized by Re Im 1 a

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

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  )=?

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  )| = | 11 22 33  H(e j  ) =  1  (  2 +  3 )

Example Re Im a dB