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【SoCVista】Noble Identities

【SoCVista】Noble Identities
【SoCVista】Noble Identities

Noble Identity

Noble Identity 是多采样率系统中最为常用的,用于将抽取器(或插值器)移动到合适的位置。如下图,以抽取器和FIR 滤波器为例,证明一下Nobel 变换的等价性,至于其他的情况可以类似得证。

假设FIR 滤波器H(z)为

1(1)011()...N N H z b b z b z ----=+++

则H(z^M)为

(1)011()...M M N M N H z b b z b z ----=+++

1) 先滤波再抽取

对x(n)用H(z^M)滤波后得到的中间值t(n)为

011()()()...((1))N t n b x n b x n M b x n N M -=+-++--

再对t(n)进行M 阶减采样,得

1011()()()()...((1))N y n t nM b x nM b x nM M b x nM N M -==+-++--

2) 先抽取再滤波

对x(n)先进行M 阶抽取得w(n)为

()()w n x nM =

再对w(n)用H(z)滤波

2011011011()()(1)...((1))

()((1))...(((1)))()()...((1))

N N N y n b w n b w n b w n N b x nM b x n M b x n N M b x nM b x nM M b x nM N M ---=+-++--=+-++--=+-++-- ----------------------------------------------------

比较y_1和y_2两个输出序列,完全一样,问题得证。

%H(z)=1+2z^(-1)+3z^(-2)+4z^(-3)

clear all

clc

t=1:0.2:10;

x=2*sin(t)./t;

%---------------------先滤波后抽取

b1=[1 0 2 0 3 0 4];

a1=[1 0 0 0 0 0 0];

y1=filter(b1,a1,x);

y1=y1(2:2:end);

%---------------------先抽取再滤波

b2=[1 2 3 4];

a2=[1 0 0 0];

x=x(2:2:end);

y2=filter(b2,a2,x);

%---------------------比较两个输出

subplot(2,1,1);

stem(y1);title('y1');

subplot(2,1,2);

stem(y2);title('y2');

抽取器的多相实现:

y(n)

图一、系统框图

上图是一个典型抗混叠抽取的系统。输入x(n)数据量很多,而抽取后的输出y(n)却要少得多,如果直接采用上图的结构,H(z)所需的计算量将是非常大的。

图二、直接实现

这是直接实现,所需乘法次数是 x(n)输入频率的N 倍,加法次数是x(n)输入频率的N-1倍。

图三、多相实现

这是低功耗的多相实现形式,与直接形式的区别是,将抽取器移动到每一个乘法器之前。

证明:假设H(z)为FIR 形式,

1(1)011()...N N H z b b z b z ----=+++

首先看看直接形式中,x(n)经过H(z)滤波后的结果w(n),如下

011()()(1)...((1))N w n b x n b x n b x n N -=+-++--

对w(n)进行M 阶抽取得到y 1(n)为

1()()y n w nM =

也就是

1011()()(1)...((1))N y n b x nM b x nM b x nM N -=+-++--

y 1(n)就是直接实现所得到的结果。下面再证明,其实y 1(n)可以用多相结构实现,问题就解决了。

仔细观察y 1(n)的表达式,与b 0相乘的只能是x(nM),x(nM)是什么?x(nM)就是x(n)的M 阶抽取啊,也就是

x(nM)

同样的道理,与b 1相乘的只能是x(nM-1),而x(nM-1)正好是x(n-1)的M 阶抽取,也可以说成x(n)延迟一个单元后的M 阶抽取

其他单元也是这个道理,看最后一项,与b N-1相乘的只能是x(nM-(N-1)),也就是x(n)

延时N-1个单元后进行M 阶抽取,如下图

这就是多相结构的来源了。。

Matlab 验证

clear all

clc

%原始24阶fir

b =[0.8147 0.9058 0.1270 0.9134 0.6324 0.0975 0.2785

0.5469 0.9575 0.9649 0.1576 0.9706 0.9572 0.4854

0.8003 0.1419 0.4218 0.9157 0.7922 0.9595 0.6557

0.0357 0.8491 0.9340];

%输入随机信号,3000点

x = rand(1,3000);

%延时x为24路

%取Q个点来观察,Q<3000/6=500

Q=30;

xd = zeros(24,Q);

for k=1:24

t = x(k:6:end);

xd(25-k,:) = t(1:Q);%取450个点来观察即可

end

w = repmat(b',1,Q).*xd;

y2 = sum(w);

%原始滤波再抽取的结果

y1 = filter(b,[1,zeros(1,23)],x);

y1 = y1(24:6:end);

y1 = y1(1:Q);

subplot(2,1,1);stem(y1);title('y1');

subplot(2,1,2);stem(y2);title('y2');

disp(max(abs(y1-y2)))

结果:y1为直接实现;y2为多相实现

比较两个序列误差为3.5527e-015,,,其实可以认为y1和y2完全相等。

英语幽默小故事10篇(带翻译)

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The owner in the middle then prepared a large sign that simply stated, "ENTRANCE". 中间战术 三个互相争生意的商店老板在一条商业街上租用了毗邻的店铺。旁观者等着瞧好戏。 右边的零售商挂起了巨大的招牌,上书:“大减价!”“特便宜!” 左边的商店挂出了更大的招牌,声称:“大砍价!”“大折扣!” 中间的商人随后准备了一个大招牌,上面只简单地写着:“入口处”。 Very Pleased to Meet You During World War II, a lot of young women in Britain were in the army. Joan Phillips was one of them. She worked in a big camp, and of course met a lot of men, officers and soldiers. One evening she met Captain Humphreys at a dance. He said to her, "I’m going abroad tomorrow, but I‘d be very happy if we could write to each other." Joan agreed, and they wrote for several months. Then his letters stopped, but she received one from another officer, telling her that he had been wounded and was in a certain army hospital in England. Joan went there and said to the matron, "I‘ve come to visit Captain Humphreys." "Only relatives are allowed to visit patients here," the matron said.

带翻译的多个英语小故事

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大学英语故事带翻译大全

大学英语故事带翻译大全 学习英语是很多英语学习者的必备之一,因为,它对英语提高有很大的帮助,以 下是小编给大家整理的关于大学英语故事带翻译大全,希望可以帮到大家关于大学英语故事带翻译大全<一> I looked at my beautiful Christmas tree and sighed. It was time. The New Year was a week old and my tree still stood in the corner of our room with its collection of memories proudly displayed in a shower of colorful lights. I'd procrastinated long enough. 我望着我的美丽圣诞树叹了口气,是时候该把树收起来了。新年已经过了一个星期,而我的圣诞树依然挺立在我们房间的角落里,在缤纷的彩灯光辉中骄傲地展示着 它珍藏其中的种种回忆。(对于收起圣诞树)我已经拖了很长时间了。 I got up, went to the garage and hauled all the boxes into the room. The garland was the first to come down. The tree looked naked already. I took the large ornaments off next. They made a large pile on our bed. An hour later, our bed was covered with Christmas memories. Each pile contained an ornament along with its matching brothers and sisters from sets purchased many years ago. 我起身去了车库,把所有的盒子都拖进了房间。花环首先被摘了下来,圣诞树看 上去就显得光秃秃的了,然后我又把大的装饰物拿了下来,它们在我们的床上堆了很 大一堆。一小时后,我们的床上堆满了圣诞回忆。每一堆都包含着一件多年前购买的 装饰物以及与之相配套的其他装饰品。 I prepared the boxes and carefully placed ornaments in their protective packaging, pausing every few minutes to admire a favorite. "Hey, little Santa!" I held the Santa from my childhood. "Thanks for being my friend for almost fifty years." He was a little ragged but still gives me a flood of wonderful memories. —Until next year, my dear friend." 我准备好盒子并把这些装饰物小心地放进包装袋内,每过几分钟就会停下来欣赏 一件我最爱的饰品。“嗨,小圣诞老人!”我拿着自童年起就陪伴我的圣诞老人,“谢谢你和我做了将近50年的朋友。”圣诞老人有一点破旧了,但仍然带给我如潮般的美好回忆,“明年再见啦,我亲爱的朋友。” There was a collection of handmade ones. My children made in their first years of school, more than twenty years ago. Made by tiny hands, they are far from perfect

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英语故事带翻译

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