EEE4176 Applications of Digital Signal Processing Fall 2011 Assistant Prof. Yangmo Yoo Dept. of Electronic Engineering Sogang University

Lecture 2: Realization of Digital Filters

Direct Realization of IIR Filters Causal, rational, stable rational function Direct realization where 3

Direct Realization of IIR Filters Canonical form Transposed form

Direct Realization of IIR Filters Direct form or direct realization (Canonical form) use minimal possible number of delays 2N+1 multipliers and 2N adders recursive realization. Transient response -> steady-state response represents delays, i.e., memory devices like latches or registers Initialization of filter : memories are initialized either to zero or to other proper values. Transposed direct realization Reverse signal flow directions in all lines Interchanging the input and the output

Parallel Realization of IIR Filters Rational transfer function : FIR + IIR ※ Note: If is complex, then is also a pole and ※ Order = (real poles) + (complex poles) where real numbers depend on

Parallel Realization of IIR Filters Ex:

Parallel Realization of IIR Filters : direct and parallel forms require the same multipliers and adders Direct form requires multiplications and additions Parallel from requires multiplications and additions # of delays are same. Better sensitivity of the frequency response to finite word length

Cascade Realization of IIR Filters Can be used to realize any IIR filters. Cascade realization Most widely used. less sensitive to quantization error The realization is not unique exist multiple ways of paring poles and zeros to make each biquads. multiple ways of ordering the biquads in the cascade connection Filtering performance can differ for different realization. Biquad 9

Cascade Realization of IIR Filters Pairing in cascade realization 1. Select the pair of complex poles nearest to the unit circles and pairs them with nearest complex zeros To make the mag respose as flat as possible Before the quantization errors accumulate too much, use poles nearest to the unit circle. 2. Continue to pair the poles and zeros according to the same rule. After complex zeros are exhausted, use real zeros 3. Pair the real poles. 10

Coupled Cascade Realization When the word length is short and filter has poles near z= + - 1 When quantization error can make the filter unstable : complex pole 11

Direct Realization of FIR Filters Standard form : N+1 Mul. N adders Transposed form Cascaded form is seldom used for FIR 12

Direct Realization of FIR FIlters Structure for linear phase FIR filters can reduce the # of multiplications by half. 13

State-space Representation State-space Concept 14

State-space Representation State equation Output equation : state vector where • State matrix and input matrix • output matrix • direct transmission matrix 15

State-space Representation For transposed direct transformation Two representations are dual or transpose to each other Note: A rational transfer function has an infinite number of state-space representations You can assign the state variables in many different ways. # of state vectors = # of memories

State-space Representation Application of state space Impulse response for and can be obtained without using z-transform and partial fraction expansion 17

State-space Representation General system response when the system is relaxed. Z-transform solution and Transfer function   18

State-space Representation Application example: Step Initialization for IIR MTI Filtering MTI(Moving Target Indicator) Filter is used for Moving target detection in RADAR, SONAR, etc. Blood flow detection in Ultrasound Imaging MTI(Moving Target Indicator) involves the detection of Doppler frequency shift from a moving object the elimination of clutter signals, which are generated by slow motions of sensors and non-target objects and MTI(Moving Target Indicator) Filtering High-pass filtering Length of an input data sequence is small Transient effect is severe. 19

State-space Representation Application example: Step Initialization for IIR MTI Filtering Transient problem Input signal can be modeled as a sum of ac signal + step function (i.e., dc offset) The response will contain the step response 20

State-space Representation Application example: Step Initialization for IIR MTI Filtering Step response 21

State-space Representation Application example: Step Initialization for IIR MTI Filtering Modified filter Conventional filter Modified filter Step response of the modified filter 22

Finite Word Length Problem Possible problems in practical filter implementation Input data has quantization error Coefficients of the difference equation are quantized Alter the pole and zero locations Intermediate data involving multiplications and additions are quantized before moving on to next step. Because of the finite word length, saturation or overflow can occur Coefficient quantization effects in digital filters Coefficient quantization affects filter response and possibly stability # of bits to represent the coefficients : For stability and 23

Finite Word Length Problem Quantized pole location in direct realization pole: High coefficient accuracy is required for a narrow band filter and high sampling rate because of the problem with the poles near 24

Finite Word Length Problem Quantized pole location in coupled realization Real and imaginary parts of the poles are independently quantized Thus the pole locations are distributed uniformly in the unit circle. Quantization of the numerator coefficients 1. Zeros of 4 common analog filters are on the imaginary axis 2. Digital filters obtained by the bilinear transformation have poles on the unit circle which are not sensitive to quantization error Sensitivity of poles Parallel or cascade realization is better than direct realization   ※ Pair the poles which are not close to each other. 25

Finite Word Length Problem Quantization effect on the frequency response Assuming the perturbations in the parameters are small, the approximate perturbation in the mag response by a first-order Taylor series is given by for a set of real parameters, to be quantized. 26

Finite Word Length Problem 27

Scaling in Fixed-point arithmetic Analog filter Saturation Polarity preserved Digital filter Overflow Reserved polarity In IIR case This causes too much problem Either or is scaled to prevent the overflow 28

Scaling in Fixed-point arithmetic Time-domain scaling where If , then 29

Scaling in Fixed-point arithmetic Frequency-domain scaling 1-norm bound Infinite-norm bound  Define : Infinite norm of : 1-norm of Note: Time-domain scaling is more conservative than freq-domain scaling  30

Scaling in Fixed-point arithmetic Scaling of Inner signals 2's complement arithmetic is preferred. Addition of all numbers does not overflow even if overflows occur in partial sums Transposed direct realization has an advantage over the direct realization Use of saturation mode is recommended   Scaling in parallel and cascade realization Read text from pp. 424-426. Quantization noise  Read text from pp. 426-436. 31