Application of digital filter in engineering —— Preprocess of test data Simple ? Complex ? Experiments or Theories ? Application and Textbook ? 2017/4/28

Original waveform 2017/4/28

Original Spectrum 2017/4/28

Frequency response of the selected filter 2017/4/28

Waveform after filtering 2017/4/28

Spectrum after filtering 2017/4/28

Digital filter 1.Filter is a particularly important class of linear time-invariant system in DSP. 2.Purpose: Filter, detection and prediction of signal. 3.Content: characteristics structures design approach. 2017/4/28

BASIC STRUCTURE OF DIGITAL FILTER CHAPTER 4 BASIC STRUCTURE OF DIGITAL FILTER h(k), k=0,1,… (Impulse Response) x(n) (Input Sequence) y(n) (Output Sequence) 2017/4/28

4.1 Representation of Digital Filter 4.2 Structure of IIR Filter CONTENTS 4.1 Representation of Digital Filter 4.2 Structure of IIR Filter 4.3 Structure of FIR Filter 4.4 Review 2017/4/28

4.1 Representation of digital filter 4.1.1 Expression of digital filter （1）Difference equation Such as: one-order system （2）Impulse response Such as: h(n) （3）System function Such as: 2017/4/28

4.1.2 Diagram （1）Block-diagram Such as: Unit Delay z-1 a0 a z-1 z-1 a1 Constant multiplication b1 Diagram of one-order digital filter Addition First-order Nth-order Block diagram 2017/4/28

Flow graph of one-order digital filter Such as: z-1 Flow graph of one-order digital filter Flow graph Different structure will determine system’s accuracy, error, stability, cost efficiency and speed, etc al. 2017/4/28

4.1.3 Implementation of the Digital Filter （1）Hardware (Special-purpose) Digital signal processor, FPGA （2）Software Digital Filter Algorithm, C, Matlab 4.1.4 Classification of the Digital Filter (1) Finite Impulse Response (FIR) FIR filter (2) Infinite Impulse Response (IIR) IIR filter 2017/4/28

Finite Impulse Response Infinite Impulse Response h(n): Finite length h(n): infinite length Nonrecursively Recursively 2017/4/28

4.2 Structure of IIR filter 4.2.1 Characteristics of IIR filter System function: At least one coefficient bi is not zero, its difference equation is: IIR filter’s structure is not unique. 具体实现结构不唯一，这话不知道对不对。 There are feedback loops in IIR structure, called recursively. There are poles inside unit circle on z-plane, otherwise unstable. IIR filter’s structure is not unique. 2017/4/28

Direct Form Cascade Form Parallel Form The same H(z) may have different implementation structure  cost efficiency computation error stability finite word-length sensitivity 2017/4/28

4.1.2 Structure of IIR filter （1）Direct Form Difference equation: System function: 直接法就是直接利用差分方程实现数字系统结构 Direct Form 2N delay units are required. 2017/4/28

（2）Direct Form II 2017/4/28

（2）Direct Form II Direct Form II N Delay Unit. z-1 z-1 x(n) y(n) b1 b2 bN a1 a2 aN z-1 a0 x(n) y(n) b1 b2 bN a1 a2 aN z-1 a0 Direct Form II N Delay Unit. 2017/4/28

We can find that: Direct Form II requires N delay units. Common disadvantage of the two form: Coefficient ai , bi can not be adjusted conveniently. When N is large, too sensitive to finite word-length  Instability  Errors. 2017/4/28

are real roots, are complex roots，and N1+N2=N （3）Cascade Form N-order transfer function can be expressed by its poles and zeros: Because H(z)’s parameter ai , bi is real value, zero ci and pole di are real roots or conjugate complex roots, that is: are real roots, are complex roots，and N1+N2=N 2017/4/28

Õ Two-order Cascade Form 表示取整 ] [ 1 · - + = z A ) ( H b a z-1 2 · - + = Õ ú û ù ê ë é N i z A ) ( H b a x(n) y(n) 11 21 11 21 z-1 1M 2M 1M 2M We can see: Cascade form’s structure is flexible, pole and zero can be adjusted conveniently and do not influence each other. Two order’s position can be selected arbitrarily, different scheme has different error and can be optimized. 2017/4/28

（4）Parallel Form Transfer function can be expressed into partial fraction, and can be composed into parallel form: expanding into real roots and conjugate complex roots form: where: N=L+2M，L, one-order network, M, two –order network. 2017/4/28

（4）Parallel Form We can see: Pole can be adjusted independently, zero cannot; If there are multi-order pole, partial fraction expansion is difficult. High speed computation, lower error than cascade form. 2017/4/28

4.3 Structure of FIR filter 4.3.1 Characteristic of FIR filter h(n) is a finite-length sequence, its system function is: FIR doesn’t have feedback loop, it is not a recursive form filter, so there is no problem of stability. 直接利用线性卷积的定义，输出y表示为输入x的延迟序列与相应的系数按位相乘的结果。 4.3.2 Structure of FIR filter （1） Direct-form x(n) y(n) h(0) z-1 h(1) h(2) h(N-1) Convolution sum 2017/4/28

（2）Cascade form Characteristics: Generally, h(n) is real value, H(z)’s zero are real or conjugate complex value.Every pair of conjugate zero composed into a second-order coefficient, so: Product of second-order factors x(n) y(n) 11 21 z-1  1M  2M 01  0M 需要更多的系数 Characteristics: Every order zero can be controlled, so can be applied when transfer zero are required to be controlled. More coefficients. More computation time. 2017/4/28

4.4 Review 4.4.1 Structure of IIR filter Recursively form, problem of stability. Pole inside the unit circle. （1）Direct form: Adjust inconvenient, sensitive to word-length effect; （2）Cascade form: Pole and zero can be adjusted independently; （3）Parallel form: Pole can be adjusted independently, low error. 4.4.2 Structure of FIR filter Non-recursively form, No stability problems. Direct form: zeros adjusted inconvenient. Cascade form: zeros adjusted convenient. 2017/4/28

Õ （4）Parallel Form (3) Cascade Form 1 + a z + a z H ( z ) = A 1 - b z é N + 1 ù ê ú ë 2 û 1 + a z - 1 + a z - 2 Õ H ( z ) = A 1 i 2 i 1 - b z - 1 - b z - 2 i = 1 1 i 2 i x(n) y(n) 11 21 11 21 z-1 1M 2M 1M 2M 2017/4/28

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