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# 信号代写|数字信号处理作业代写digital signal process代考|Number Representation

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## 信号代写|数字信号处理作业代写digital signal process代考|Fixed-point Number Representation

In general, an arbitrary real number $x$ can be approximated by a finite summation
$$x_{Q}=\sum_{i=0}^{w-1} b_{i} 2^{i}$$
where the possible values for $b_{i}$ are 0 and 1 .
The fixed-point number representation with a finite number $w$ of binary places leads to four different interpretations of the number range (see Table $2.1$ and Fig. 2.35).

The signed fractional representation (2’s complement) is the usual format for digital audio signals and for algorithms in fixed-point arithmetic. For address and modulo operation, the unsigned integer is used. Owing to finite word-length $w$, overflow occurs as shown in Fig. 2.36. These curves have to be taken into consideration while carrying out operations, especially additions in 2’s complement arithmetic.

## 信号代写|数字信号处理作业代写digital signal process代考|Floating-point Number Representation

The representation of a floating-point number is given by
$$x_{Q}=M_{G} 2^{E_{G}}$$
with
$0.5 \leq M_{G}<1$,
where $M_{G}$ denotes the normalized mantissa and $E_{G}$ the exponent. The normalized standard format (IEEE) is shown in Fig. $2.40$ and special cases are given in Table 2.3. The mantissa $M$ is implemented with a word-length of $w_{M}$ bits and is in fixed-point number representation. The exponent $E$ is implemented with a word-length of $w_{E}$ bits and is an integer in the range from $-2^{w_{E}-1}+2$ to $2^{w_{E}-1}-1$. For an exponent word-length of $w_{E}=8$ bits, its range of values lies between $-126$ and $+127$. The range of values of the mantissa lies between $0.5$ and 1 . This is referred to as the normalized mantissa and is responsible for the unique representation of a number. For a fixed-point number in the range between $0.5$ and 1, it follows that the exponent of the floating-point number representation is $E=0$. To represent a fixed-point number in the range between $0.25$ and $0.5$ in floatingpoint form, the range of values of the normalized mantissa $M$ lies between $0.5$ and 1 , and for the exponent it follows that $E=-1$. As an example, for a fixed-point number $0.75$ the floating-point number $0.75 \cdot 2^{0}$ results. The fixed-point number $0.375$ is not represented as the floating-point number $0.375 \cdot 2^{0}$. With the normalized mantissa, the floating-point number is expressed as $0.75 \cdot 2^{-1}$. Owing to normalization, the ambiguity of floating-point number representation is avoided. Numbers greater than 1 can be represented. For example, $1.5$ becomes $0.75 \cdot 2^{1}$ in floating-point number representation.

## 信号代写|数字信号处理作业代写digital signal process代考|Effects on Format Conversion and Algorithms

First, a comparison of signal-to-noise ratios is made for the fixed-point and floatingpoint number representation. Figure $2.43$ shows the signal-to-noise ratio as a function of input level for both number representations. The fixed-point word-length is $w=16$ bits. The word-length of the mantissa in floating-point representation is also $w_{M}=16$ bits, whereas that of the exponent is $w_{E}=4$ bits. The signal-to-noise ratio for floating-point representation shows that it is independent of input level and varies as a sawtooth curve in a $6 \mathrm{~dB}$ grid. If the input level is so low that a normalization of the mantissa due to finite number representation is not possible, then the signal-to-noise ratio is comparable to fixed-point representation. While using the full range, both fixed-point and floating-point result in the same signal-to-noise ratio. It can be observed that the signal-to-noise ratio for fixed-point representation depends on the input level. This signal-to-noise ratio in the digital domain is an exact image of the level-dependent signal-to-noise ratio of an analog signal in the analog domain. A floating-point representation cannot improve this signal-tonoise ratio. Rather, the floating-point curve is vertically shifted downwards to the value of signal-to-noise ratio of an analog signal.

## 信号代写|数字信号处理作业代写DIGITAL SIGNAL PROCESS代考|FIXED-POINT NUMBER REPRESENTATION

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