2-7 FUNDAMENTALS OF ANALOG-TO-DIGITAL CONVERSION AND VICE-VERSA
`
`25
`
`Sampling
`
`Analog
`domain
`
`Continuous-valued
`Sampling
`continuous-time
`signals
`
`Reconstruction
`
`ADC
`
`Quantization
`
`Continuous-valued
`discrete-time
`signals
`
`Digital
`domain
`
`010010, 011001, ...
`
`Discrete-valued
`discrete-time
`signals
`
`Voltage mapping
`
`Discrete-valued
`continuous-time
`signals
`
`DAC
`
`Figure 2-15: Data conversion between the analog and digital domains. Analog signals are converted into digital by
`sampling and quantization. Digital signals are converted into analog by mapping numbers into voltage levels, followed
`by reconstruction filtering.
`
`In other words,
`
`[δ( f − f0) + δ( f + f0)].
`
`A 2
`
`A cos(2π f0t)
`
`The PSD is
`
`A2
`4
`
`SX ( f ) =
`
`[δ( f − f0) + δ( f + f0)] ,
`and the average power is
`
`2-7 Fundamentals of
`Analog-to-Digital Conversion
`and Vice-Versa
`
`This section covers the conversion of analog to digital
`signals, and vice-versa. There are two parts of this
`conversion:
`
`(1) sampling a continous-time signal to a discrete-
`time signal, which takes on continuous values at
`times that are rational numbers, and
`
`(2) quantization of the continuous values that the
`signal
`takes on into values that are rational
`numbers. The quantized and sampled signal is
`called a digital signal. This is illustrated in
`Fig. 2-15.
`
`Some sources produce analog signals, such as
`speech and audio. Conversion between analog and
`digital representations utilize the fundamental oper-
`ations illustrated in Fig. 2-15. An analog-to-digital
`converters (ADC) converts analog signals, which are
`continuous in both time and amplitude,
`to digital
`
`Px =Z ∞
`
`SX ( f ) d f =
`
`A2
`2
`
`.
`
`−∞
`The autocorrelation function is the inverse Fourier
`transform of the PSD, which is straightforward to
`determine:
`
`Rx(τ) =
`
`A2
`2
`
`cos(2π fcτ).
`
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`

`26
`
`CHAPTER 2 SOURCES OF INFORMATION
`
`Buffer
`
`Hold
`capacitor
`
`Figure 2-17: Block diagram of a sample-and-hold
`circuit.
`
`is to keep the input signal
`purpose of the circuit
`constant during the ADC conversion. Their input is
`the analog signal to be sampled and the output is
`a piecewise constant signal. While there are many
`different implementations, all S/H circuits include a
`switching circuit, a holding capacitor, and an output
`buffer which is an amplifier with a gain equal to one.
`A block diagram of a sample-and-hold circuit is
`shown in Fig. 2-17. There could also be an input buffer
`(not shown) to supply the necessary current to charge
`the hold capacitor. When the switch closes, the circuit
`samples the input signal. Ideally the time to sample
`is infinitely small. When the switch opens, the S/H
`circuit operates in hold mode. The output buffer has
`a very high input impedance and the capacitor does
`not discharge; i.e., the capacitor holds the voltage. The
`circuit remains in hold mode for the duration of the
`sampling period Ts. Therefore, the impulse response
`of an ideal S/H circuit is
`
`(2.120)
`
`(cid:19) ,
`
`1 2
`
`h(t) = u(t)− u(t − Ts) = Π(cid:18) t
`Ts −
`
`and the frequency response is
`
`H( f ) = Ts
`
`sinπ f Ts
`π f Ts
`
`e− jπ f Ts .
`
`(2.121)
`
`Suppose that the input to the sampler is the analog
`signal x(t) with a corresponding Fourier transform
`X ( f ), and the output of the sampler is the sampled
`signal xs(t). The sampled signal can be viewed in the
`time domain as a product between x(t) and a sequence
`
`signals, which are discrete in both time and amplitude.
`A digital-to-analog converter (DAC) converts a digital
`representation into an analog signal and can be thought
`of as the inverse of the ADC. The analog signal can
`then be used to drive a speaker or other analog output
`device.
`An ADC typically requires three subsystems—an
`anti-aliasing filter (AAF), a sampler, and a quantizer
`(Fig. 2-16). Each one of these three subsystems is
`examined next.
`
`AnƟ-aliasing
`filter
`
`Sample/hold
`
`QuanƟzer
`
`Figure 2-16: High-level block diagram of an ADC.
`
`2-7.1 Sampling
`
`Sampling a signal x(t) at a sampling period Ts is
`described by
`
`x[nTs] = x(t)(cid:12)(cid:12)t=nTs
`
`.
`
`(2.118)
`
`Sampling can also be represented as an inner
`product with a Dirac delta function at the sampling
`instant t = nTs:
`
`(2.119)
`
`x[nTs] = hx(t), δ(t − nTs)i
`=Z ∞
`x(t) δ(t − nTs) dt.
`
`−∞
`The main parameter of a sampler is the sampling
`frequency fs = 1/Ts. The time between two samples
`is the sampling period Ts. Samplers take regularly
`separated analog measurements of the signal’s ampli-
`tude. Most samplers operate on voltage, but some can
`operate on current. While conceptually, sampling is
`performed instantaneously, in practice it can take a
`small but nonzero amount of time to “sample.” Most
`samplers return the average amplitude during the time
`they actively “sample.”
`Sampling is typically performed by electronic
`circuits called sample-and-hold (S/H) circuits. The
`
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`

`2-7 FUNDAMENTALS OF ANALOG-TO-DIGITAL CONVERSION AND VICE-VERSA
`
`27
`
`where the coefficients xn are equal
`products
`
`to the inner
`
`xn =* ∞
`∑
`n=−∞
`ZT
`δ(t) e− j2πnt/T dt =
`
`1 T
`
`=
`
`δ(t − nT ), e− j2πnt/T+
`
`(2.127)
`
`.
`
`1 T
`
`Therefore, the Fourier series of the impulse train is
`
`e− j2πnt/T .
`
`(2.128)
`
`∞∑
`
`1 T
`
`δ(t − nT ) =
`
`∞∑
`
`n=−∞
`n=−∞
`From the property given by Eq.
`(2.43) with
`f0 = n/T , it follows that e j2πtn/T
`δ( f − n/T ). In
`conclusion, there are two Fourier transform represen-
`tations:
`
`e− j2π f nT ,
`
`∞∑
`
`δ( f − n/T ).
`(2.129)
`
`∑n
`
`1 T
`
`n=−∞
`
`
`
`
`δ(t − nT )
`
`∞∑
`
`n=−∞
`
`Therefore the Fourier transform of an impulse train
`with period T is another impulse train with period
`T −1 Hz.
`Using the Fourier transform of a product is the con-
`volution of the Fourier transforms (see Appendix A).
`Because in the time domain the sampled signal given
`by Eq. (2.122) is a product, its spectrum is given by
`the convolution
`
`s(cid:19)
`
`n T
`
`∞∑
`
`n=−∞
`
`1 T
`
`s
`
`n T
`
`δ(cid:18) f −
`Xs( f ) = X ( f )∗
`X(cid:18) f −
`s(cid:19)
`n=−∞
`[··· + X ( f + fs) + X ( f ) + X ( f − fs) +··· ]
`
`∞∑
`
`1 T
`
`s
`
`1 T
`
`s
`
`=
`
`=
`
`X ( f − n fs) .
`
`(2.130)
`
`∞∑
`
`n=−∞
`n6=0
`
`= fs X ( f ) + fs
`
`of Dirac delta functions:
`
`(2.122)
`
`δ(t − nTs).
`
`∞∑
`
`xs(t) = x(t)
`
`n=−∞
`The function ∑n=∞
`n=−∞ δ(t − nT ) is called an impulse
`train or a sampling function.
`Because the product x(t) δ(t − nTs) is zero every-
`where except at the sampling instances nTs, x(t) can
`be replaced with the discrete time signal x[nTs] without
`changing the result:
`
`x[nTs] δ(t − nTs).
`
`(2.123)
`
`∞∑
`
`n=−∞
`
`xs(t) =
`
`Example 2-11:
`
`Fourier transform of a
`pulse train
`
`Determine the Fourier transform of the impulse train
`∑n=∞
`n=−∞ δ(t − nT ). This will be used in the derivation
`of the sampling theorem below.
`
`Solution: From Appendix A, the Fourier transform
`relationship of the Dirac delta function is
`
`δ(t − nT )
`
`e− j2π f nT ,
`
`since
`
`Z ∞
`
`δ(t − nT ) e− j2π f t dt = e− j2π f t(cid:12)(cid:12)t=nT = e− j2π f nT .
`
`(2.124)
`
`−∞
`Because the Fourier transform is linear,
`
`e− j2π f nT .
`
`(2.125)
`
`∞∑
`
`n=−∞
`
`δ(t − nT )
`
`∞∑
`
`n=−∞
`
`The above result leads to another useful relation-
`ship. The signal ∑∞
`n=−∞ δ(t − nT ) is periodic and can
`be represented using a Fourier series with e− j2πnt/T as
`basis functions:
`
`xne− j2πnt/T ,
`
`(2.126)
`
`∞∑
`
`n=−∞
`
`δ(t − nT ) =
`
`∞∑
`
`n=−∞
`
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`

`28
`
`CHAPTER 2 SOURCES OF INFORMATION
`
`From this equation we can conclude that the spec-
`trum of the sampled signal is periodic with a period
`in frequency equal to the sampling frequency
`fs = 1/Ts (Fig. 2-18). In the frequency domain, there
`are infinitely many replicas of the spectrum of the
`continuous-time signal. These replicas are frequency-
`translated versions of the spectrum of the original
`signal. When these replicas overlap,
`the effect
`is
`known as aliasing. Sampling maps analog frequencies
`in the range [0, ∞) to discrete-time frequencies in
`the range [0, fs/2). The frequency fs/2 is called the
`Nyquist frequency.
`Let us consider in more detail the sampling of a
`real-valued signal. Recall that a real-valued signal is
`absolutely bandlimited if there exists an fmax such
`that X ( f ) = 0 for frequencies | f| > | fmax|. Figure
`2-18 illustrates the spectrum of a real-valued signal,
`where the signal is absolutely bandlimited to fmax.
`It also shows the spectrum of the signal sampled at
`fmax > fs/2 (so that there is aliasing), but the sampling
`frequency is greater than the bandwidth of the signal:
`
`(2.131)
`
`< fmax < fs.
`
`fs
`2
`If there is aliasing, some higher-frequency compo-
`nents of the analog signal appear “on top” of some
`of the lower-frequency components. Only one com-
`ponent is formed as a result; i.e., the higher-frequency
`components become indistinguishable from the lower-
`frequency components and cannot be separated by
`filtering later. In the aliasing example in Fig. 2-18(b),
`the components from fs/2 to fmax overlap with the
`components from fs/2 down to fs − fmax.
`The presence of aliasing over part of the spectrum
`does not mean that the entire signal is useless. If the
`signal of interest is in the bandwidth [0,
`fs − fmax],
`aliasing can be acceptable since the aliased part
`of the spectrum is not of interest. In Fig. 2-18(b),
`over the band [0,
`fs − fmax] there is no aliasing and
`this portion can be used and processed further. For
`example, the portion with aliasing can be filtered out
`with a low-pass digital filter so as to preserve the
`band [0,
`fs − fmax]. However, if fs ≤ fmax, aliasing
`will extend over the entire bandwidth of the signal,
`rendering the sampled signal useless.
`
`The replicas of the spectrum just touch each other if
`fs = 2 fmax, which still results in aliasing. The replicas
`of the spectrum do not overlap as long as fs > 2 fmax,
`as illustrated in Fig. 2-18(c). The minimum sampling
`frequency to avoid aliasing completely is
`
`fs = 2 fmax
`
`(2.132)
`
`and is known as the Nyquist sampling frequency.
`It is important to recognize that if the signal is not
`bandlimited, sampling always introduces aliasing. For
`some signals the aliasing error is always significant.
`For example,
`the Dirac delta function cannot be
`sampled;
`i.e., the discrete-time delta function (the
`Kronecker delta) cannot be obtained by sampling.
`Another function that cannot be sampled is the step
`function. However, for signals that are effectively
`bandlimited most of the energy is in a finite band and
`the aliasing error can be made small by choosing the
`sampling frequency to be sufficiently large.
`One objective in the design of an ADC is to
`minimize the effect of aliasing. If the signal
`to
`be sampled is not bandlimited or its spectrum is
`unknown, aliasing is generally minimized by inserting
`an anti-aliasing filter (AAF), in front of the ADC,
`as was illustrated earlier in Fig. 2-16. The AAF is
`designed to remove all frequency components above
`fs/2; i.e., the AAF is ideally a lowpass filter with a
`passband from 0 to fs/2 Hz.
`
`2-7.2 Sampling a complex-valued signal
`
`In the above discussion on sampling, we assumed
`that the signal being sampled is a real-valued signal,
`having a one-sided bandwidth W = fmax. Recall from
`Section 2-8 that the bandwidth of complex-valued
`signals is double-sided. A complex-valued signal is
`bandlimited if there exists a band [ f1, f2] such that
`X ( f ) = 0 outside [ f1, f2]. In particular, consider a
`complex-valued signal with double-sided bandwidth
`W = f2 − f1 (and where f1 is negative), as was
`illustrated earlier in Fig. 2-11. Then, to completely
`avoid aliasing, the sampling frequency should be
`
`fs ≥ W = f2 − f1
`
`(2.133)
`
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`

`2-7 FUNDAMENTALS OF ANALOG-TO-DIGITAL CONVERSION AND VICE-VERSA
`
`29
`
`– fmax
`
`fmax
`
`(a) Spectrum X( f )
`
`s– f
`
`f
`s
`
`– f
`
`max
`
`fmax
`
`sf
`
`(b) Aliased spectrum
`
`s– f
`
`– f
`
`max
`
`0
`
`f
`
`max
`
`fs
`
`(c) Not-aliased spectrum
`
`Figure 2-18: (a) Spectrum of a continuous-time real-valued signal, (b) spectrum of the discrete-time signal obtained
`after sampling the signal in (a) at fmax < fs < 2 fmax, and (c) spectrum of the discrete-time signal obtained after sampling
`the signal in (a) at fs ≥ 2 fmax.
`
`or equivalently,
`
`(2.134)
`
`fs + f1 ≥ f2.
`Figure 2-19 illustrates the spectrum of a sampled
`complex-valued signal, where the sampling frequency
`is chosen according to Eq. (2.134), so that aliasing is
`avoided.
`Aliasing is only partial; i.e., there is a frequency
`band that is still useful, as long as
`− fs + f2 < fs + f1.
`In other words, so long as the sampling frequency is
`greater than one-half of the bandwidth of the signal,
`fs > ( f2 − f1)/2 = W /2,
`
`(2.135)
`
`(2.136)
`
`then there is no aliasing over the frequency band from
`− fs + f2 to fs + f1; aliasing is limited to the band from
`fs + f1 to f2 (Fig. 2-20).
`If there is aliasing, some components at negative
`frequencies appear ”on top” of and indistinguishable
`from components at positive frequencies. The portion
`with aliasing can be filtered out with a low-pass digital
`filter that will preserve the band [− fs + f2, fs + f1].
`However, if fs ≤ W /2 aliasing will extend over the
`entire bandwidth of the signal.
`Sampling complex-valued signals is referred to as
`quadrature sampling. Each of the real and imaginary
`components occupies only one-half of the bandwidth
`
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`

`30
`
`…
`
`…
`
`f
`
`– fs
`
`f1
`
`f2
`
`fs – f1
`
`fs
`
`Figure 2-19: Spectrum of a sampled complex-valued
`signal with no aliasing.
`
`…
`
`…
`
`f
`
`– fs
`
`f1
`
`f2
`
`fs
`
`fs – f1
`
`Figure 2-20: Spectra of a sampled complex-valued
`signal with some aliasing.
`
`of the complex-valued signal. Therefore, quadrature
`sampling reduces the required sampling rate by a
`factor of two at the expense of having to use two ADCs
`instead of one. This is illustrated in Fig. 2-21.
`
`I
`
`Q
`
`ADC
`
`ADC
`
`Figure 2-21: A quadrature ADC performs the
`analog-to-digital conversion of the in-phase (I) and
`quadrature (Q) components separately.
`
`2-7.3 Bandpass sampling
`
`Nyquist sampling theory requires the sampling fre-
`quency to be at least twice the maximum frequency
`
`CHAPTER 2 SOURCES OF INFORMATION
`
`of the signal to avoid aliasing. This means that the
`passband of the anti-aliasing filter (AAF) should ex-
`tend from 0 to fs/2, which is appropriate for lowpass
`signals. Some signals of interest are bandpass in
`nature; i.e., they are centered around some frequency
`fc > 0 with one-sided bandwidth W . If the sampling
`frequency is required to be at least twice the highest
`frequency component of bandpass signals, then the
`sampling frequency must be high. This makes the cost
`and power consumption of the ADC and the digital
`signal processing high. To sample continuous-time
`bandpass signals it is advantageous to use a technique
`called bandpass sampling.
`By way of an example, consider a real-valued signal
`centered at 70 MHz with a bandwidth of 20 MHz. The
`highest frequency component is 70 + 20/2 = 80 MHz.
`According to Nyquist’s sampling theory, it should be
`sampled at a sampling frequency of at least 160 MHz.
`Suppose, however, that this signal is sampled with a
`sampling frequency of 40 MHz (twice the bandwidth
`of the bandpass signal). The periodic spectrum of the
`sampled signal is shown in Fig. 2-22. We see that the
`replicas of the spectrum just run into each other, which
`still shows aliasing. However, it motivates us to find
`ways to sample at rates below 160 MHz and avoid
`aliasing completely. This is indeed possible if both of
`the following conditions are true:
`
`and
`
`fs >
`
`2 fc + W
`n + 1
`
`=
`
`2 fU
`n + 1
`
`fs <
`
`2 fc − W
`n
`
`=
`
`2 fL
`n
`
`.
`
`(2.137a)
`
`(2.137b)
`
`Here, fL and fU are the lower and upper band-edges,
`and n is an arbitrary positive integer with a maximum
`fL/W . Lowpass sampling corresponds to
`value of
`n = 0, in which case Eq. (2.137a) becomes fs < ∞.
`With bandpass sampling, aliasing can be avoided
`if the sampling frequency is only larger than twice
`the bandwidth of the bandpass signal. This approach
`clearly has important advantages such as allowing
`the use of a slower, and therefore, potentially less
`expensive ADC.
`To understand bandpass sampling, let us return to
`
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`2-7 FUNDAMENTALS OF ANALOG-TO-DIGITAL CONVERSION AND VICE-VERSA
`
`31
`
`–120 –100 –80 –60 –40 –20 0 20 40 60
`
`80 100 120
`
`f (MHz)
`
`(a) fs = 40 MHz (n = 1)
`
`–120 –100 –80 –60 –40 –20 0 20 40 60
`
`80 100 120
`
`f (MHz)
`
`(b) fs = 55 MHz (n = 2)
`
`–120 –100 –80 –60 –40 –20 0 20 40 60
`
`80 100 120
`
`f (MHz)
`
`(c) fs = 100 MHz (n = 3)
`
`Figure 2-22: Bandpass sampling of a real-valued signal centered at 70 MHz with a bandwidth of 20 MHz with a
`sampling frequency equal to (a) 40 MHz, (b) 55 MHz), and (c) 100 MHz.
`
`the example in Fig. 2-22. Since fL/W = 3, n should be
`less than 3. There is a range of sampling frequencies
`that can be used to avoid aliasing. If n = 1, aliasing is
`avoided if both conditions
`
`and
`
`fs > 80 MHz
`
`(2.138a)
`
`fs < 120 MHz
`
`(2.138b)
`
`are satisfied at the same time. If n = 2, the require-
`ments are
`
`and
`
`fs > 53.33 MHz,
`
`(2.139a)
`
`fs < 60 MHz.
`
`(2.139b)
`
`For n = 3, the requirements are
`
`fs > 40 MHz,
`
`fs < 40 MHz,
`
`(2.140a)
`
`(2.140b)
`
`which is not possible, so aliasing is unavoidable
`(although borderline) for n = 3. In practice,
`it
`is
`desirable to separate the spectrum replicas by filtering.
`If the copies of the spectrum run up to each other, this
`separation becomes impossible with a real filter, which
`leads to the requirement that the sampling frequency
`be chosen in a way that allows a certain guard band.
`Figure 2-22 shows the spectrum when the sampling
`frequency is 100 MHz (n = 1) and 55 MHz (n = 2),
`in addition to the borderline aliasing case when the
`sampling frequency is 40 MHz.
`
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`32
`
`CHAPTER 2 SOURCES OF INFORMATION
`
`We note that when n = 1, the spectrum of the
`located between DC and fs/2,
`sampled signal,
`is
`inverted, compared with the spectrum of the bandpass
`signal over the positive frequency axis. This is called
`spectrum inversion, and it is a result of frequency
`translation where the negative frequency component
`appears at positive frequencies. Spectrum inversion
`is a proof
`that negative frequencies do,
`in fact,
`exist, and that sampling and frequency translation are
`inseparable. All odd values of n make the negative
`frequency component appear between 0 Hz and
`the Nyquist frequency fs/2 and result in spectrum
`inversion.
`Spectrum inversion is only a problem if it is not
`known whether or not it has occurred, which can
`happen if knowledge of n is not available once a signal
`is sampled. If it is known that spectrum inversion has
`occurred, it can be corrected by simple digital signal
`processing (see Problem 6.4 at the end of Chapter 6).
`Using an even value of n avoids spectrum inversion
`altogether.
`
`the input signal of an ADC is always specified, for
`example, 1 V or 2 V peak-to-peak.13
`A key parameter of the quantization process is the
`number of bits, which we will denote as b, and the
`number of quantization levels 2b. In general, these
`quantization levels may or may not be uniformly
`distributed. Figure 2-23 shows uniformly distributed
`quantization levels that are also symmetric with
`respect to zero. Such a quantizer is referred to as a
`mid-rise quantizer, as its output can never be exactly
`zero. (If we were to attempt to make one quantization
`level identically zero, then the characteristic cannot be
`symmetric since the number of quantization levels is
`even.)
`Quantization is always associated with an error.
`Quantization error occurs because in the quantiza-
`tion process, the observed amplitude value is either
`rounded up or rounded down. Quantization can be
`mathematically modeled as a process that adds an
`error e[n] to the input signal x[n]:
`
`xq[n] = e[n] + x[n],
`
`(2.142a)
`
`2-7.4 Quantization
`
`or
`
`(2.142b)
`
`e[n] = xq[n]− x[n].
`Assuming that clipping is avoided, the quantization
`error can be modeled as a random variable uncorre-
`lated with the input signal.
`If quantization is uniform, the quantization step
`size is equal to the least significant bit (LSB) of the
`quantized signal, which is also equal to the range of
`the quantizer divided by the number of quantization
`levels:
`
`(2.143)
`
`V 2
`
`b ,
`
`∆ =
`
`or V = 2b∆. Under these assumptions (no clipping and
`uniform quantization) the quantization error has a PDF
`that is uniform within the range [−∆/2, ∆/2]:
`f (x) =(1/∆ −∆/2 ≤ x ≤ ∆/2,
`
`0
`
`otherwise.
`
`(2.144)
`
`13We assume without loss of generality that the signal being
`quantized is a voltage signal. For a current signal the analysis
`would be similar.
`
`The third element in the block diagram of the ADC in
`Fig. 2-16 is the quantizer. The quantizer converts the
`constant value provided by the sampler x[nTs] into a
`digital number:
`
`xq[n] = Q [x[nTs]] .
`
`(2.141)
`
`Quantizers are characterized by three main pa-
`rameters: (1) the range - the difference between the
`maximum and minimum values they can quantize,
`(2) the number of bits, and (3) the distribution of the
`quantization levels.
`A signal outside the range of the quantizer will
`be clipped, resulting in nonlinear distortion, which is
`generally severe and unacceptable. To avoid having to
`deal with the nonlinear effects of clipping, we will
`assume that the input signal is limited to the range
`[−V /2,V /2]; i.e., the full-scale range of the quantizer
`is V . The quantization range V is chosen so that the
`probability that a sample x[nTs] might exceed the range
`−V /2 to V /2 is negligible. This is why the range of
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`2-7 FUNDAMENTALS OF ANALOG-TO-DIGITAL CONVERSION AND VICE-VERSA
`
`33
`
`-V/2
`
`7Δ/2
`
`5Δ/2
`
`3Δ/2
`
`Δ/2
`
`Output
`
`-Δ/2
`
`-3Δ/2
`
`-5Δ/2
`
`-7Δ/2
`
`V/2
`
`Input
`
`Figure 2-23: Quantization levels of a symmetric mid-rise quantizer.
`
`(PAPR) of the signal as14
`
`Ppeak
`Pave
`The peak power of a signal x(t) is Ppeak = max|x(t)|2.
`
`η =
`
`.
`
`(2.148)
`
`For example, for the single tone signal
`
`A cos(2π f t + θ),
`
`the peak power is A2, the average power is A2/2 (see
`Eq. (2.23)) and the PAPR is equal to A2/(A2/2) = 2.
`We assume that the signal being quantized occupies
`the entire range of the quantizer; i.e., it is in the range
`of [−V /2,V /2]. Its peak power is
`2(cid:19)2
`
`=(cid:18)2b ∆2(cid:19)2
`Ppeak =(cid:18)V
`= 22(b−1)∆2
`
`(2.149)
`
`and the average power is
`
`Pave =
`
`Ppeak
`η
`
`14Instead of the PAPR, sometimes a parameter called crest
`factor is used, defined as the square root of the PAPR.
`
`Therefore, the average of the quantization error is zero
`(which is the next best thing to actually not making an
`error in the first place). With a simple derivation, the
`variance, or the power of the quantization noise, can
`be shown to be
`
`∆2
`12
`
`=
`
`V 2
`12(22b)
`
`.
`
`(2.145)
`
`x2 dx =
`−∆/2
`The quantization error is unavoidable, but its variance
`can be minimized by increasing the number of bits.
`Rather than the power of the quantization noise, a
`more useful metric is the ratio between the average
`signal power and the quantization noise power, called
`SQNR:
`
`Z ∆/2
`
`1 ∆
`
`σ2
`q =
`
`SQNR =
`
`Pave
`σ2
`q
`
`.
`
`(2.146)
`
`In decibels
`
`SQNR (dB) = 10 log10 Pave − 10 log10 σ2
`q .
`
`(2.147)
`
`We can define the peak-to-average power ratio
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`

`34
`
`CHAPTER 2 SOURCES OF INFORMATION
`
`Therefore, measured in dB, the signal-to-quantization-
`noise ratio is
`
`and since Ppeak = V 2/4, the PAPR is η = 3 = 4.77 dB.
`The signal-to-quantization noise ratio then becomes
`
`SQNR = 10 log10
`
`Ppeak
`ησ2
`q
`
`= 10 log10
`
`= 10 log10
`
`22b∆212
`4η∆2
`3× 22b
`η
`= 20b log10 2 + 10 log10 3− 10 log10 η dB
`= 6.02b + 4.77− 10 log10 η dB.
`(2.150)
`The greater the number of bits used, the greater the
`precision of the digital representation and the greater is
`the resulting SQNR. If the number of bits is increased
`by one, the SQNR for (b + 1) bits is
`
`SQNR(b + 1) = 6.02(b + 1) + 4.77− 10 log10 η dB
`= SQNR(b) + 6.02b
`dB.
`(2.151)
`
`Therefore, each additional quantizer bit improves the
`SQNR by approximately 6 dB, which is known as the
`6 dB per bit rule of thumb.
`It is obvious that to actually calculate the SQNR
`we need to know the average signal power, which
`can be estimated for several special cases. Since
`every periodic signal can be represented as a sum of
`harmonic components, an important special case is the
`SQNR for a cosine signal x(t) = A cos(2π f0t + θ). If
`it is assumed that this signal occupies the full range
`of the quantizer without clipping, A = V /2, and the
`average power (see Eq. (2.23)) is one-half of the peak
`power, or η = 2 or 3 dB, then the SQNR is
`
`SQNR = 6.02b + 4.77− 10 log 10 2 = 6.02b + 1.76 dB.
`(2.152)
`Another special case is when the input signal is
`random with a uniform distribution over the full range
`of the quantizer [−V /2,V /2]. Then the power of the
`input signal is (see Eq. (2.145)):
`
`Pave =
`
`V 2
`12
`
`,
`
`SQNR = 10 log10
`
`V 2/12
`V 2/(12× 22b)
`= 10 log10 22b = 6.02b
`
`dB.
`
`(2.153)
`
`Note that if the signal x(t) does not use the full range of
`the quantizer, then the SQNR will be proportionately
`less. In other words, the SQNR depends linearly on the
`power of the input signal, so long as the input signal
`does not cause clipping.
`The output of quantization is some binary represen-
`tation of the input. Numbers can be represented using
`floating point or fixed point arithmetic. Fixed-point
`numbers with m integer bits including the sign bit and
`n fractional bits can be represented as
`
`(xm−1xm−2 . . . x0x−1 . . . x−n)2.
`In other words there is a radix point somewhere in
`the middle of the digits. The position of the radix
`point can be implied and not explicit. The notation
`(··· )b means a base-b number. For binary numbers
`xk ∈ {0, 1}. Fixed-point binary numbers can be
`represented using sign-magnitude, one’s complement,
`or two’s complement format.
`In sign-magnitude representation, the leftmost bit
`represents the sign and the remaining bits represent
`the magnitude; i.e., the absolute value. If the number
`is negative the most significant bit is set to 1.
`The one’s complement representation of a positive
`number is its binary representation. The one’s com-
`plement of a negative number is obtained by first
`writing the positive value of the number in binary
`arithmetic and then inverting all the bits; i.e., swapping
`0s for 1s, and vice versa. One’s complement was used
`in the early history of computing, but most modern
`computers use two’s complement.
`The two’s complement of a negative number is
`formed by taking the two’s complement of the corre-
`sponding positive number; i.e., subtracting the positive
`number from 2. Alternatively, the two’s complement
`representation can be obtained by complementing
`(i.e. inverting) every bit (except the sign bit or the
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`2-7 FUNDAMENTALS OF ANALOG-TO-DIGITAL CONVERSION AND VICE-VERSA
`
`35
`
`most significant bit) and then adding 1 to the least
`significant bit.
`to decimal
`To convert from two’s complement
`representation, the following formula can be used:
`
`Compression
`
`Uniform
`quantization
`
`Expansion
`
`Figure 2-24: Nonuniform (logarithmic) quantization
`can be performed by compression,
`followed by
`standard uniform quantization and expansion.
`
`For nonuniformly distributed signals, nonuniform
`quantization is more appropriate. One simple way
`to implement nonuniform quantization is to use a
`compandor (compressor-expander). According to this
`technique the input signal
`is first processed by a
`nonlinear element called a compressor which provides
`more gain to lower-amplitude input values, but leaves
`large amplitude values unchanged. Then a uniform
`quantizer is used, followed by a block with the reverse
`nonlinearity, called an expander (Fig. 2-24).
`Because the expander implements the reverse non-
`linearity,
`the product of the characteristics of the
`compressor and expander is equal to 1.
`In practice, compressors and expanders with loga-
`rithmic characteristics are used. According to the so-
`called µ-law, the transfer function of the compressor
`is
`
`F(x) = sgn(x)
`
`,
`
`(2.154)
`
`ln (1 + µ|x|)
`ln (1 + µ)
`
`where µ = 127 for 7-bit quantization or 255 if 8-bit
`quantization is used. The range of x(t) is assumed to
`be between −1 and 1. The inverse relationship is
`h(1 + µ)|y| − 1i .
`F−1(y) = sgn(y)
`
`(2.155)
`
`1 µ
`
`Since the 1970s, this µ-law has been adopted in the
`telephone networks of the USA, Canada, and Japan.
`A-law with another logarithmic characteristic is used
`
`x−k2−k.
`
`n∑k
`
`=1
`
`xk2k +
`
`m−2
`
`∑k
`
`=0
`
`x = −xm−12m−1 +
`
`Therefore the dynamic range of x is
`
`x ∈(cid:2)−2m−1 ··· 2m−1 − 2−n(cid:3) .
`
`2-7.5 PCM
`
`The term pulse code modulation (PCM) is really
`a misnomer, but has remained in use. PCM is not
`a modulation method, but simply analog-to-digital
`conversion. PCM was introduced in the 1970s to
`denote sampling a toll-quality voice signal at a rate of
`8000 samples/s, followed by quantization using 8 bits
`per sample. The justification for the term PCM is that
`the voice signal is represented using a binary ”code”
`in preparation for digital transmission.
`Since narrowband speech extends approximately
`over the range from 300 Hz to 3.4 kHz, prior to digi-
`tization, the bandwidth of the voice signal is limited
`by an anti-aliasing filter to that range, allowing the
`sampling rate to be 8000 Hz. Using 8 bits per sample,
`according to PCM a voice signal is encoded into a 64
`kilobits per second (which we will abbreviate as kb/s)
`digital data stream for transmission. This sampling
`and quantization is according to the International
`Telecommunication Union (ITU) standard G.711.
`More generally, sampling any signal of bandwidth
`W using a sampling frequency fs > 2W , followed by
`quantization with b bits of resolution to achieve a data
`rate of 2bW is being called PCM.
`Note that SQNR is an indication of the quality of
`the quantized signal. As discussed earlier, assuming
`uniform quantization, the SQNR is proportional to the
`signal power. For speech, this means that low signal
`values will experience low quality. Perceptually this
`is not desirable. Uniform quantization is appropriate
`when the input signal is uniformly distributed over
`the entire range of the ADC [−V /2,V /2]. However,
`speech signals do not fall in that category.
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`36
`
`CHAPTER 2 SOURCES OF INFORMATION
`
`in the rest of the world, according to which
`
`and
`
`Xs( f )
`
`x[nTs] e− j2πn f Ts .
`
`(2.160)
`
`∑n
`
`1 f
`
`s
`
`1 f
`
`s
`
`X ( f ) =
`
`=
`
`The analog signal x(t) is the inverse Fourier transform
`of X ( f )
`
`x(t) =Z ∞
`
`−∞
`
`X ( f ) e j2π f t d f
`
`x[nTs] Z fmax
`− fmax
`sin[2π fmax(t − nTs)]
`π(t − nTs)
`x[nTs] sinc [2 fmax(t − nTs)] . (2.161)
`
`x[nTs]
`
`∑n
`
`e j2π f (t−nTs) d f
`
`∑n
`
`∑n
`
`1 f
`
`s
`
`1 f
`
`s
`
`2 fmax
`fs
`
`=
`
`=
`
`=
`
`Assuming that fs = 2 fmax,
`
`x(t) = ∑
`
`n
`
`x[nTs] sinc ( fst − n) .
`
`(2.162)
`
`The above result is very similar to the discrete-time
`linear convolution defined as
`
`y[m] = ∑
`
`l
`
`h[l] x[m− l].
`
`(2.163)
`
`Therefore, Eq. (2.162) represents convolution between
`the discrete-time sequence x[nTs] and the impulse
`response of an ideal
`lowpass filter as given by
`Eq. (2.63), h(t) = sinc(2 fmaxt), namely
`
`(2.164)
`
`x(t) = x[nTs]∗ h(t − nTs).
`The difference from Eq. (2.163) is that the output is
`a continuous-time function; i.e., Eq. (2.164) describes
`an LTI system with discrete input and analog output.
`Such an LTI system can be built, as illustrated in
`Fig. 2-25. The first step produces a signal that is
`continuous-time, but discrete in amplitude, and then
`in the second step, the actual analog signal is obtained
`
`,
`
`1 A
`
`|x| ≤
`
`1 A
`
`A|x|
`1 + ln A
`1 + ln (A|x|)
`1 + ln (A)
`
`F(x) = sgn(x)
`
`
`≤ |x| ≤ 1.
`(2.156)
`where A = 87.6. The performance of µ-law and A-law
`is very similar. µ-law achieves slightly higher dynamic
`range at the cost of slightly larger distortion for small
`signals. The µ-law and A-law models were designed
`specifically so that the SQNR is relatively constant and
`does not depend on the signal power. Via a complex
`derivation, it can be shown that the SQNR for µ-law
`companding is approximately
`
`(2.157a)
`
`SQNR = 4.77 + 6b− 20 log10[ln(1 + µ)]
`≈ 6b− 10 dB.
`The PCM standard describing µ-law and A-law,
`known as G.711, was established in 1972 and has been
`in use ever since. It is still implemented at present by,
`for example, all Voice over Internet Protocol (VoIP)
`platforms.
`
`(2.157b)
`
`2-7.6 Reconstructing an analog signal from
`discrete-time samples:
`digital-to-analog conversion
`
`The reverse process of analog-to-digital conversion
`is digital-to-analog conversion. To investigate how a
`real-valued analog signal can be reconstructed, we will
`assume that x(t) has been sampled without aliasing;
`i.e., x(t) is bandlimited to a one-sided bandwidth fmax
`that is less than fs/2 Hz. Then, over