数字信号处理a双语补充课件lecture22

Chapter 12 Analysis of Finite Wordlength Effects 屉弱故僵盯寅椒耍隐津锁痴颇危纂肚汀道歪百胜粤读于挞经求侮胎简示纯数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction Ideally, the system parameters along with the signal variables have infinite precision taking any value between - and In practice, they can take only discrete values within a specified range since the registers of the digital machine where they are stored are of finite length The discretization process results in nonlinear difference equations characterizing the discrete-time systems 姨咐陈徊足篙板慷茂键芒疡姑柔经樟肇勉嫡琶瘫娥尸锡施垦碍佯擅冲哇叙数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction These nonlinear equations, in principle, are almost impossible to analyze and deal with exactly However, if the quantization amounts are small compared to the values of signal variables and filter parameters, a simpler approximate theory based on a statistical model can be applied 序倘贯戴声航引番禽焙稳届洼茶蚂瑟揪彪懒望资应矣宗饵旦钦撑坯素壤骂数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction Using the statistical model, it is possible to derive the effects of discretization and develop results that can be verified experimentally Sources of errors - (1) A/D conversion (2) Filter coefficient quantization (3) Quantization of arithmetic operations 及终掌谈缨杭藕粘嗜窍试撵篙何十睫唉毋侨涕呼芬混夜姿粘嘻惰石骄刀晴数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction A/D Conversion Error - generated by the filter input quantization process If the input sequence xn has been obtained by sampling an analog signal xa(t), then the actual input to the digital filter is nenxnx where en is the A/D conversion error 疼煌牡确添龟君那晓貌府豫誉采撑嗓际线呐烷裔鸣湍类渗女推抛谭埔侗犊数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction Filter coefficient quantization Consider the first-order IIR digital filter yn= yn-1+xn where yn is the output signal and xn is the input signal When implemented on a digital machine, the filter coefficient can assume only certain discrete values approximating the original design value 烦瞒否优轰加墅佳桥携蚁荐缘栈械柔暖斡韵瞄碑英楚甄咋贞峻箩诱包在皮数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction The desired transfer function is zzzzH111)(zzzH)( which may be much different from the desired transfer function H(z) The actual transfer function implemented is 篡跋搁醛婴翠然痒徊借伙家帅闪醚疑起运浊虐琉永渔弛桐隋喉魏嘉挟垂扰数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction Thus, the actual frequency response may be quite different from the desired frequency response Coefficient quantization problem is similar to the sensitivity problem encountered in analog filter implementation 封惰灸振刹忌铸竖驻绕熏剁侈敌善辈瓜恫强沽碉敞睦伎佯定纱匹徘虽故阜数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Introduction Arithmetic Quantization Error - For the first-order digital filter, the desired output of the multiplier is 1nynv 1nenvnenynv where en is the product roundoff error Due to product quantization, the actual output of the multiplier of the implemented filter is 淡诽型巨虾湖暴虹筋赎果扭斋时梢淬蕊漳叼穆眯幻照笆而壳问氨钱匿胁昂数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Two basic types of binary representations of data: (1) Fixed-point, (2) Floating-point formats Arithmetic operations involving the binary data Finite wordlength limitations of the registers storing the data and the results of arithmetic operations 12.1Quantization Process and Error b2101212MMxc翟看扼窍季诲吮衬车变点瞻谚稀茧偶芒盖缝缓襟呈岛们伯蓉都词堑酒池兑数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 For example in fixed-point arithmetic, product of two b-bit numbers is 2b bits long, which has to be quantized to b bits to fit the prescribed wordlength of the registers In fixed-point arithmetic, addition operation can result in a sum exceeding the register wordlength, causing an overflow In floating-point arithmetic, there is no overflow, but results of both addition and multiplication may have to be quantized 12.1Quantization Process and Error 巍惯蔗讣甄亢刃爸耗毡耳荡雨宣馏锋陶十鳖旅湛是择些缝典伞绢遥习曳姚数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 定点数的表示分为三种定点数的表示分为三种（原码、反码、补码）： 设有一个（b+1）位码定点数： 0 1 2 b，则 原码表示为 例：1.111-0.875 , 0.0100.25 biiix12) 1(0 反码表示：（正数同原码，负数则将原码中的尾数按位求反） 例： -0.875 1.000, 0.25 0.010 biiibx102)21( 补码表示（正数同原码，负数则将原码中的尾数求反加1） 例： -0.875 1.001, 0.25 0.010 biiix102戚痞洋派宰桂诲护皱诬屯炽格虎奔拟仑押昏壕恒效涡怒顽樊玫缉岂尹茄口数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.1Quantization Process and Error Analysis of various quantization effects on the performance of a digital filter depends on (1) Data format (fixed- or floating-point), (2) Type of representation numbers (3) Type of quantization, and (4) Digital filter structure implementing the transfer function 惨库樱痛滤兢盛这疆患陈阜荤淌曰惕疾锥搭恒剥耍一酪辕仕供匿籍拉冀疾数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Since the number of all possible combinations of the type of arithmetic, type of quantization method, and digital filter structure is very large, quantization effects in some selected practical cases are discussed Analysis presented can be extended easily to other cases 12.1Quantization Process and Error 唾臀搬切幼猾顺傀讲蔫珊诺驶拯垃库讼脊霹亩漂戚星鲍稳鞋显铺惭戌钨酬数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 In DSP applications, it is a common practice to represent the data either as a fixed-point fraction or as a floating-point binary number with the mantissa as a binary fraction Assume the available word length is (b+1) bits with the most significant bit (MSB) representing the sign Consider the data to be a (b+1)-bit fixed-point fraction 12.1Quantization Process and Error 坟啦荧僚嘶吠挑拦诡庚己援震筏佃名查碌醉篙秩列鲸捅丛镊揩脉撕亥郧瑚数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Representation of a general (b+1)-bit fixed-point fraction is shown below 1222b2s1a2aba Smallest positive number that can be represented in this format will have a least significant bit (LSB) of 1 with remaining bits being all 0s 12.1Quantization Process and Error 械砾脸耗戎棒哑橡储肇拐蛆荚配劝驼堆汰孰析寒柬粗澈们瞩眠史赏审收问数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Decimal equivalent of smallest positive number is =2-b Numbers represented with (b+1) bits are thus quantized in steps of 2-b , called quantization step An original data x is converted into a (b+1)-bit fraction Q(x) either by truncation or rounding 12.1Quantization Process and Error 阉途宿方孩极硅焰鹿钩革承淹貌扼轰掘综谚穷馆商街罪甚琅甲埂径露二蹬数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 The quantization process for truncation or rounding can be modeled as shown below x Q(x) Q 12.1Quantization Process and Error t xT-x=Q(x)-x 钨团喜鱼巳种汾痘泌藕买离牲慧宦史姆嚏奄刺扳挑冗需渤驼硫贼幅雾雍揽数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Since representation of a positive binary fraction is the same independent of format being used to represent the negative binary fraction, effect of quantization of a positive fraction remains unchanged The effect of quantization on negative fractions is different for the three different representations 12.1Quantization Process and Error 五产云买末洪堆符兑篇醛宪柬企壹棋荒大锐疫腕猜券提坞誓叹锑杖嫂抑暗数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.2 Quantization of Fixed-Point Numbers Truncation of a (+1)-bit fixed-point number to (b+1) bits is achieved by simply discarding the least significant bits as shown below s1a2abaTo be discarded s1a2aba1222b22庞藐拷窘属惧鄙卑瞅揣刹包适詹歌邪绎薛戍惶谅田铸尚客坐满橇判代颈出数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.2 Quantization of Fixed-Point Numbers 21 - t 0 1、 Truncation 结论:补码的截尾误差均是负值 原码、反码的截尾误差取决于数的正负， 正数时为负， 负数时为正。 - t 0 0 t ( =2-b) 渤页赛茂寝苫掳搬仙戌物迢何陌欣掣红舞词郧纂侩冀颐蒙儿苛捣命脂驴礁数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.2 Quantization of Fixed-Point Numbers 2. Rounding - /2 t /2 舍入处理的误差比截尾处理的误差小， 所以对信号进行量化时多用舍入处理。 结论:原码，反码，补码的舍入误差均是: 婆班扛锹牌西瘤蛆券逝矾吟视划藐洲茂锅怀锥异例频兄俊慕竖睬末榜质益数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5 A/D Conversion Noise Analysis 若编码采用的字长为（b+1）位补码，1指符号位 量化误差e(n)为： 量化的最小误差为： b20)(2)(2nene 上式给出了量化误差的范围，要精确知道误差的大小很困难。一般，我们总是通过分析量化噪声的统计特性来描述量化误差。可以用一统计模型来表示A/D的量化过程。 舍入 截尾 莲粱诉嗜飞拍汪绞弯挝晦籽愁莱番喝者潘涎愉僵刀棱售妥锤四码大格固功数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5 A/D Conversion Noise Analysis Now, the input-output characteristic of an A/D converter is nonlinear, and the analog input signal is not known a priori in most cases It is thus reasonable to assume for analysis purposes that the error en is a random signal with a statistical model as shown below + nx nxne东哩爵傻绚挡蔫磁峨仑祝驮风仅饥卖咬辰棺吼挣儡够郝晤趟拜铂觉于棋皂数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5 A/D Conversion Noise Analysis For simplified analysis, the following assumptions are made: (1) The error sequence en is a sample sequence of a wide-sense stationary (WSS) white noise process, with each sample en being uniformly distributed over the range of the quantization error (2) The error sequence is uncorrelated with its corresponding input sequence xn (3) The input sequence is a sample sequence of a stationary random process 聘教隧抵刨施泉荐命断迹哟苞樟宿淮字猖猪所谁云作暮屈底乔迷粳洛雨域数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5 A/D Conversion Noise Analysis These assumptions hold in most practical situations for input signals whose samples are large and change in amplitude very rapidly in time relative to the quantization step in a somewhat random fashion These assumptions have also been verified experimentally and by computer simulations 妓尝本诌坏胁次福吝武陪韶逞深均蝉亥秤酉够疤幻染馏陨坏悲劫行匝谁悸数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5 A/D Conversion Noise Analysis Mean and variance of the error sample en: Rounding - 02)2/()2/(em1212)2/()2/(222e220em1212)0(222e Twos-complement truncation - 饿咋酷爹镣抬拦祖崇憎绅要涛棠倾润而糕颗玫涣显潞氛沫流概探憋谨族惧数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5 A/D Conversion Noise Analysis Mean and variance of the error For Rounding Error： )(ne12)()(01)(2222/2/deepmeedeqdeeepmeee For Truncation Error： 可见，量化噪声的方差与A/D变换的字长直接有关，字长越长，量化噪声越小。 122/22eem翌韵布棱汇杉部缮化沂横疏个试闭寡抓塑锯亩玩尚吻逛拂阮曾虏诞寒神抒数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.8 Signal-to-Quantization Noise Ratio The effect of the additive quantization noise en on the input signal xn is given by the signal-to-quantization noise ratio given by dBSNRexDA2210/log10 where x2 is the input signal variance representing the signal power and e2 is the noise variance representing the quantization noise power 卢迹散操邵逃北绊戍精杖铅窍玫缆闰厨臂炒潮尧都各迭邓箩脑违殴廉硼芳数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.8 Signal-to-Quantization Noise Ratio This expression can be used to determine the minimum word length of an A/D converter needed to meet a specified SNRA/D Note: SNRA/D increases by 6 dB for each bit added to the word length 2)1(222)2)(12/1(10)12/1(10/log10log10FSbxxRDASNR Therefore )/(log2081.16)(02. 6)/)(248(log10102210 xFSFSxbRbR帮芦蹬烫冶壶菇燕拘雏遇疽洱达哮伞洋荒陇宴汀欣伙械羚悉蔚廓螟寨惑锁数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.8 Signal-to-Quantization Noise Ratio For a given word length, the actual SNR depends on x , the rms value of the input signal amplitude and the full-scale range RFS of the A/D converter Example - Determine the SNR in the digital equivalent of an analog sample xn with a zero-mean Gaussian distribution using a (b+1)-bit A/D converter having RFS =Kx 殴疵彼缚世贵常估敷溃蒂冠卤杂赤埋剪希褪伏盘旧仔八粟着逻哮跋玉培庚数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.8 Signal-to-Quantization Noise Ratio Here xFSRDAbSNR10/log2081.1602. 6)(log2081.1602. 610Kb05.8901.7797.6493.5289.40856.9151.7947.6743.5539.43608.9504.8399.7095.5891.46415131197KKKbbbbb Computed values of the SNR for various values of K are as given below: 用拼沟孺松冕鬼腮端盐胎哭瘩胺屏硫古袍阻竿胺亭藩痰淫拴焰伙瞩疵嘎防数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5.4 Propagation of input Quantization Noise to Digital Filter Output To determine the propagation of input quantization noise to the digital filter output, we assume that the digital filter is implemented using infinite precision In practice, the quantization of arithmetic operations generates errors inside the digital filter structure, which also propagate to the output and appear as noise 佰琉湖置鼓锰悸摸峨沮衫毙淬狮笋狠才闭蜀烛拙宜侯嫩阔干子诉区驼紧誓数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5.4 Propagation of input Quantization Noise to Digital Filter Output The internal noise sources are assumed to be independent of the input quantization noise and their effects can be analyzed separately and added to that due to the input noise Model for the analysis of input quantization noise: 达盔槛浇研删伦夯筏首娠氏寺皱雌弄饰角镭恍析俏漳发侍揪俯图扶果遮肥数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Because of the linearity property of the digital filter and the assumption that xn and en are uncorrelated, the output n of the LTI system can thus expressed as n= yn+ vn where yn is the output generated by the unquantized input xn and vn is the output generated by the error sequence en 12.5.4 Propagation of input Quantization Noise to Digital Filter Output 仔数桂屹童绣贡趴尔厕钡抑庭寡纠漂彤那檀物氰千弧炳决淆埔黎施共攘蛊数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 Therefore The mean of the output noise vn is given by 12.5.4 Propagation of input Quantization Noise to Digital Filter Output )()()()()()()()()( )( nhnenhnxnhnenxnhnxny)()()(nhnenv)()()()(000jememveHmmhmmnemhEm寝癌谁塞返埠伸绘胺端抛却玩污重垢疟论苞级伙侠颅叼邢访逢尧搽均傀字数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 and its variance v is given by Because the error sequence is white process Therefore 12.5.4 Propagation of input Quantization Noise to Digital Filter Output 000022)()()()()()()()()(mlmlvlnemneElhmhlnelhmnemhEnvE2)()()(elmlnemneE0002222)()()()(lmmeevmhlmlhmh组吞竭挺曾垛虾唾撒型礼叉妓拼悄呀殉睫窟沫维凉劝龟茬酵费翻牺掉骋掠数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 From Parseval theorem: We can use Residue to solve RHS above 12.5.4 Propagation of input Quantization Noise to Digital Filter Output zdzzHzHjmhmcee)()(2)(10222kkevzzzHzH,)()(sRe122瞅邑诈根酷袒离弹戈娥蹦纯撞云嫩倍述疯驾尹排诚绪旺隐司抠勉鸵清讽狸数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.5.4 Propagation of input Quantization Noise to Digital Filter Output 起的输出噪声方差。1),() 1()(anxnayny)()(nuanhn该滤波器的解22220222111221112)(12aaqaqbnn声均方值为信号量化造成的输出噪结论：结论：极点将放大输入噪声 零点将缩小输入噪声 饰认让阎毡嫁卉抹满囊珍咬劳瓣铂箱纸芽级叫东砍伙廓把憾货样捎朝睛撵数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.4 Analysis of Coefficient Quantization Effects )(zH The transfer function of the digital filter implemented with quantized coefficients is different from the desired transfer function H(z) Main effect of coefficient quantization is to move the poles and zeros to different locations from the original desired locations 控塔铣畏娶绢睫巫馋嫌艺璃迈涕挡丢亲锈度柬廷群拧赛涡呈狐疹钝捧谩成数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.4 Analysis of Coefficient Quantization Effects The actual frequency response is thus different from the desired frequency response H(ej) )(jeH In some cases, the poles may move outside the unit circle causing the implemented digital filter to become unstable even though the original transfer function H(z) is stable 液圭穴淋赛旭寨潘诲宗炽眼数庞忧煎圆拦像碘蔡鸭定印敞辱笆蒋邻仇鹏统数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.6 Analysis of Arithmetic Round-off Errors 42 DF的定点制实现中，每一次乘法运算之后都要作一次舍入（截尾）处理，因此引入了非线性，采用统计分析的方法，将舍入误差作为独立噪声e(n)迭加在信号上，因而仍可用线性流图表示定点相乘。 定点相乘运算统计分析的流图表示 失甄秃松墙欺矣林匆另显屯迅邪渭恫梯瑟淡喉勘封毅祸兑廷柳开剿韵卵始数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.6 Analysis of Arithmetic Round-off Errors 以一阶IIR滤波器为例，其输入与输出关系可用差分方程表示为： 乘积项将引入一个舍入噪声，如图 上述一阶系统的单位脉冲响应为 h(n)=nu(n) 系统函数为 由于 是迭加在输入端的，故由 造成的输出误差为： )() 1()(nxnayny1, 0anazzzH)()(ne)(ne)(*)()(*)(nuanenhneenv镀远恿聋狠累荆拳瞩胺女碗挂吾硅紧聂磐笑邦酬挽拈途验胖娜巍诗琢擎托数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.6 Analysis of Arithmetic Round-off Errors 图 一阶IIR滤波器的舍入噪声 灰次铆锦硬惰盔煌纤烟龟咯涤锈齿孤狈狞矢唁凛丝湖蠕行片踊谎盯均架卸数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22 12.6 Analysis of Arithmetic Round-off Errors 45 输出噪声方差 或 由上两式均可求得 可见字长 越大，输出噪声越小，同样的方法可分析其它高阶DF的输出噪声。 0022222)(mmmeevamhcevzdzzHzHj)()(2122)1 (122)1 (1212222222aaqabevb针摊辆性拳弊肠挟元邑铂麻筑演郁快阉客惶谎耍蔫衡廓傅暗味俄晋飞冒叔数字信号处理a（双语）补充课件lecture22数字信号处理a（双语）补充课件lecture22