at different frequencies, called frequency division duplexing (FDD), a duplexing filter separates them from each other. A duplexing filter consists of two bandpass filters. If the transmitter and receiver operate in different time slots, called time division duplexing (TDD), a switch can be used to isolate the receiver during transmission. The loss of a switch is usually less than the loss of a duplexing filter.

A high level of integration is essential in the mass production of lowcost transceivers and receivers. The ultimate goal is to integrate all the transceiver electronics on a single chip, because external components increase the cost. High-quality filters and resonators are often too difficult to integrate on a chip. The practical solution is to combine several chips made using different technologies with some external components. Low-cost siliconbased technologies such as bipolar, BiCMOS, and CMOS technology are used to produce chips up to about 3 GHz. When top performance is required or the frequency is higher, chips are made using GaAs and InP technologies.

11.2 Noise

Random fields and voltages, that is, noise, disturb all radio systems. The antenna receives noise from its surroundings, and all receiver components, which are either active or lossy, generate noise. We call the former the antenna noise and the latter the receiver noise; their sum is called the system noise. In a radio system (e.g., a communication link) the system noise power in the receiver bandwidth may be stronger than the signal to be received. The ratio of the signal power to the noise power at the receiver bandwidth, that is, the S /N often determines the quality of a radio link. However, noise signals may also be useful, as is the case in radiometry, for example, in remote sensing and radio astronomy (see Sections 12.7 and 12.8).

In system considerations, a radio channel, where white noise corrupting the signal is the only nonideality, is called the additive white Gaussian noise (AWGN) channel. In addition to noise, in practical radio channels there are other nonidealities. When the small-scale fading or Rayleigh fading in multipath propagation conditions is the limiting factor for the channel performance, we call it a Rayleigh fading channel.

11.2.1 Receiver Noise

In a receiver, many kinds of noise are generated, for example, thermal noise, shot noise, 1/f noise, and quantum noise.

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Thermal noise is generated by the thermal motion of charge carriers. The warmer the material is, the more electrons collide with the crystal lattice of the material. Each collision causes a change in the kinetic energy state of the electron, and the energy difference is radiated as an electromagnetic wave. Similarly, collisions are also the reason for resistivity of a material and, therefore, thermal noise is generated in all materials and circuits absorbing RF power. Thermal noise is directly proportional to the absolute temperature of the medium, but its power density is independent of frequency—it is socalled white noise.

Shot noise is often the most important noise mechanism in semiconductor devices and electron tubes. Shot noise is caused by the fact that charge is not a continuous quantity but always a multiple of an electron charge. For example, a current going through the Schottky interface is not continuous but is a sum of the current impulses of single electrons. The power density of shot noise is directly proportional to the current.

At low frequencies there is 1/f noise (flicker noise) in all semiconductor devices. It is caused, for example, by the fluctuating amount of electrons in the conduction band. Its power density is inversely proportional to frequency.

Quantum noise is due to the quantized energy of the radio wave. It is important only in cases of submillimeter and shorter waves, because their energy quantum W = hf is large.

Noise properties of a device are described by the noise factor F or the equivalent noise temperature Te [4]. The latter is also called the effective input noise temperature, the input noise temperature, or just the noise temperature. The noise factor of a linear two-port is defined [5] by

where Nin is the available noise power in a bandwidth df from a matched resistive termination (here ‘‘matched’’ means that the termination is matched to the characteristic impedance of the line) at temperature T0 = 290K connected to the input of the device, and Nout is the total noise power available at the output port in a bandwidth df when the input power is Nin . Ga is the available power gain of the two-port for incoherent signals from an input bandwidth of df to an output bandwidth of df . The noise factor indicates how many times larger the output noise power of the device is compared to that of a noiseless device, when both have in the input a matched resistive termination at the absolute reference temperature of T0 = 290K.

The equivalent noise temperature is defined by means of the noise factor as follows:

or

In other words, the equivalent noise temperature can be defined as the physical temperature, at which the matched resistive input termination of a noiseless device should be in order to have the same available noise power in the output as the noisy device itself produces into its output when its matched resistive input termination is at the absolute zero temperature.

A resistor R at temperature T generates noise, the rms voltage of which in a bandwidth of df is

where h = 6.626 × 10−34 Js is Planck’s constant and k = 1.381 × 10−23 J/K is Boltzmann’s constant. The approximation of (11.4) is valid when hf << kT. The available noise power from this resistor, that is, the noise power from this resistor to another resistor with the same resistance, is

This noise power P is equal to Nin in (11.1), and therefore the noise factor is

In (11.6) Nout is sometimes for practical reasons the power delivered (coupled) to the load; then, instead of the available power gain, one must use the transducer power gain Gt (see (5.26)). Because the available noise

where Gt , max is the maximum value of the transducer power gain. In respect to white noise in the input, the device behaves like a device that has a gain of Gt , max over the noise bandwidth Bn and a gain of zero at all frequencies outside this band.

Let us next consider the equivalent noise temperature of a resistive attenuator at a physical temperature T. If both the attenuator itself and its matched resistive input termination are at the same physical temperature, the attenuator absorbs and emits the same amount of energy. When we measure the noise power from this attenuator in a bandwidth of df , we get the following result (assuming hf << kT ) as in (11.5)

Nout = kdf T = Nin e −t + Nint = kdf Te −t + kdf T (1 − e −t )

(11.11)

where t is the so-called optical depth, which describes the rate of absorption and emission in the attenuator. For a section of a transmission line with a length l and an attenuation constant a, the optical depth is t = al , and therefore the attenuation of this attenuator is L = e al = e t. Now we know that the intrinsic noise power Nint is

By the definition of the equivalent noise temperature, the input noise temperature TL of a resistive attenuator at a physical temperature T is then

A receiver is, from the signal and noise point of view, all the way from the antenna terminals to the detector a chain of linear two-ports, as shown in Figure 11.5. The noise factor of such a chain is

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Figure 11.5 Chain of two-ports in series.

In (11.14) and (11.15) Fi , Ti , and Gi are the noise factor, noise temperature, and available gain of the i th two-port of the chain. Equations (11.14) and (11.15) are called the Friis noise equations. They are valid only if the twoports are matched to each other.

According to (11.15), the first stage determines the noise temperature of a receiver, if G1 is high enough and T2 low enough. If possible, a good amplifier should be placed as the first stage of the receiver. In a comparison of different amplifiers, a quantity called the noise measure is used. The noise measure is

which is the noise factor of an infinitely long chain of similar amplifiers minus one. When cascading two amplifiers, it is better to put as the first stage an amplifier, which has the lowest noise measure, and not necessarily the one, which has the lowest noise factor.

Example 11.1

You have two low-noise amplifiers, LNA1 and LNA2, with characteristics T1 = 100K, G1 = 13 dB, and T2 = 90K, G2 = 7 dB, respectively. You want to use these LNAs together in a series connection in the input of a lownoise receiver. Which one should be placed as the first stage in order to obtain the best possible noise performance of the receiver?

Solution

Let us first calculate the corresponding noise factors: F1 = 1 + T1 /T0 = 1 + 100/290 = 1.34, and F2 = 1 + 90/290 = 1.31. Gains of the amplifiers in absolute values are G1 = 20.0 and G2 = 5.0. Now we can calculate the noise measures: M 1 = (F1 − 1)/(1 − 1/G1 ) = 0.34/0.95 = 0.36, and M 2 = 0.31/0.8 = 0.39. Therefore, LNA1 should be placed as the first stage.

The noise factor and noise temperature of a mixer are quantities, which continuously cause confusion, especially in the case of millimeterwave receivers, where a mixer is often the first stage [6]. The reason for the confusion is the presence of an image sideband in the mixer. The most frequent errors in the noise factor and noise temperature usage are made in the following areas:

•The SSB quantities of a DSB mixer (also called a broadband mixer ) are confused with the respective quantities of an SSB mixer (also called a narrowband or image-rejection mixer );

•Depending on the situation, the noise generated in the image termination is to be included as a part of either the receiver noise or the source noise;

•Many old rules of thumb, valid for calculating SSB quantities from DSB quantities of a DSB mixer (or vice versa) in a special case, are unfortunately used also in other cases where they are not valid.

In a mixer, power is converted to the intermediate frequency not only from the signal sideband but also from other sidebands, especially from the image sideband. Let us consider a DSB mixer, which has conversion losses L s and L i from the signal and image sideband, respectively, to the intermediate frequency band, and the conversion loss values from other (harmonic) sidebands are infinitely large. The DSB noise temperature TM , DSB of this mixer is, according to the definition of the equivalent noise temperature, the

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temperature of a termination that is connected to the noiseless mixer at both the signal and image sidebands. However, the SSB noise temperature TM , SSB is the temperature of a termination, according to the definition that is connected to the noiseless mixer only at the signal sideband, and at the image sideband there is a termination at the temperature of 0K. Then the DSB and SSB noise temperatures of the DSB mixer are related to each other as

where L c is the conversion loss of the mixer, TM is the mixer noise temperature, and TIF is the noise temperature of the IF amplifier. The receiver SSB

noise temperature is obtained with TM = TM , SSB and L c = L M , SSB = L s , and the DSB noise temperature is obtained with TM = TM , DSB and L c = L M , DSB = L s L i /(L s + L i ). For diode mixers L c is larger than unity and, therefore, the noise temperature of the IF amplifier plays a very important

role in the receiver noise temperature.

It is worth emphasizing that if an image-rejection filter is placed in front of the DSB mixer, the mixer turns into an SSB mixer, and its conversion loss L c is no more equal to the original L s neither is its noise temperature the one obtained from (11.17).

So far we have assumed that when we use a DSB mixer in an SSB mode, there is a termination at 0K at the image sideband. However, this is not the case in practice. For example, a radiometer observing the atmospheric molecular lines often utilizes a DSB mixer as its first stage, and although the useful signal now enters the receiver only at the signal sideband, the same atmospheric background noise enters the receiver at both sidebands, and the termination impedance at both sidebands is nearly the same, that is, the antenna radiation impedance. In this case we have to add a term (L s /L i ) Ti into the receiver noise temperature of (11.18) in order to have a noise quantity, which really describes the receiver’s ability to detect the useful signal at the signal sideband. Here Ti is the image sideband termination temperature, which is often nearly equal to the antenna noise temperature (see Section 11.2.2), that is, Ti ≈ TA .

Now we are ready to summarize the receiver noise temperatures and noise factors in the case of a DSB mixer used as the first stage [6]:

Example 11.2

Calculate the receiver noise temperature and noise figure of a receiver consisting of an LNA, followed by an image-rejection mixer, and an IF amplifier.

The characteristics of the components are TLNA = 50K and GLNA = 10 dB, TM = 500K and L M = 6 dB, TIF = 200K and GIF = 50 dB. How much worse is the receiver performance if at a room temperature of 295K there

is a cable with a loss of 0.2 dB in front of the LNA?

Solution

The component gains in absolute values are GLNA = 10, L M = 4, and GIF = 100,000, but the latter is not needed in this calculation. TR = TLNA +

(1/GLNA ) TM + (L M /GLNA ) TIF = 50K + (1/10)500K + (4/10)200K =

(50 + 50 + 80)K = 180K. FR (dB) = 10 log (1 + TR /T0 ) = 10 log 1.62 = 2.1 dB. When the cable with loss of L = 0.2 dB (in absolute value L is 1.05)

is added in front of this receiver, we get a new receiver noise temperature

TR′ = (L − 1) Troom + LTR = (1.05 − 1)295K + 1.05 × 180 = 15K + 189K = 204K.

When the aim is to have a very low-noise receiver, the receiver front end is cooled down to, for example, a temperature of 20K. The cooling reduces the noise temperature of an amplifier or a mixer but also the thermal noise from resistive loss in transmission lines and other components. Some