1. Radio Receiver Systems
Chapter 1
Radio Receiver Systems
1.1 Receiver Design
The input to a wireless transmitter may be voice, video, data, or other information to be
transmitted to one or more distant receivers. These data are usually referred to as the
baseband signal. A basic function of the transmitter is to modulate the baseband
information onto a high frequency sine wave carrier signal for the purpose of efficient
transmission over a noise-filled radio channel, than the direct radiation of the baseband
signal.
The purpose of a radio frequency (RF) receiver is to process incoming energy into useful
information while maintaining a sufficient signal-to-noise ratio (SNR), adding a minimum
of distortion. This must be done for widely varying RF power level and in the presence of
noise and interferers.
Performance depends:
- system design
- internal circuitry
- operating environment
Receiver performance specifications:
- high gain to restore the lower power of the received signal to a level near its original
baseband value.
- receiver sensitivity quantifies the ability of a receiver to respond to weak signal
levels.
Analogue receivers – maximum RF level to ensure a certain demodulated S/N ratio.
Digital receivers – maximum BER at a certain RF level as a measure of
performance.
- receiver selectivity usually refers to a receiver’s ability to reject unwanted signals on
adjacent channel frequencies. Typical ranges from 70 to 90dB which is difficult to
achieve that many systems do not allow simultaneous active adjacent channels in
the same geographical area.
- spurious response rejection is required as all receivers have the potential for
responding to frequencies other than the desired channel. Minimised by the proper
choice of the IF and use of RF filters. Spurious response rejection of 70 to 100dB
can be achieved in practical receivers.
- isolation from transmitter to avoid saturation of the receiver.
- intermodulation rejection measures the receiver tendency to generate its own onchannel interference from two or more strong off-channel signals. 70dB is easily
achievable.
- receiver self-quieting refers to reduced receiver sensitivity on some channels due to
internally generated signals that capture the detector and thus prevent or inhibit the
reception of desired, weak signal.
Other receiver parameters prominent in many regulatory requirements:
Distortion of the demodulated signal
S/N ratio for strong signals
Frequency stability, cochannel rejection, cross modulation, radiated emission,
susceptibility to high RF levels
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1. Radio Receiver Systems
1.2 Receiver Architecture
The radio signal experiences many transformations in its migration from a baseband
signal at the transmitter to its replication at the receiver. Primary motivation for selecting
certain receiver architecture is the required performance.
It may be instructive to walk through Fig.1.1 block by block and summarise the important
decisions for each block before tackling the somewhat iterative procedure for determining
individual stage specifications.
Antenna:
receives electromagnetic waves radiated from many sources over a relatively broad
frequency range and transfer the energy through the input circuits
Filter 1 or the preselector: a highly selective, cavity tuned filter, cascaded with a lowpass filter:
to limit the bandwidth of spectrum before reaching the RF amplifier and mixer in
order to minimise IM distortion
to attenuate receiver spurious response
to suppress local oscillator energy originating in the receiver
RF amplifier or low noise amplifier (LNA):
to amplify the possibly very weak received signal while minimising the noise power
that is added to the received signal
to attenuate local oscillator energy
to isolate filter 1 from filter 2 so that the overall selectivity is not destroyed
Filter 2 or image filter:
to attenuate receiver spurious response frequencies
to attenuate direct IF frequency pickup
to attenuate noise at the image frequency originating in or amplified by the RF
amplifier
to suppress second harmonics originating in the RF amplifier
to suppress local oscillator energy leaking back into the antenna
1st mixer: active, passive, unbalanced, singly balanced, doubly balanced, tuned,
broadband
needs to have high intercept point
to down-convert the received RF signal to a lower frequency signal called the
intermediate frequency (IF)
Injection filter:
to attenuate wideband noise around the LO frequency and its harmonics
to attenuate the second harmonics in order not to degrade mixer second-order
intercept point (IP2)
1st local oscillator:
its single sideband (SSB) phase noise determine the receiver’s adjacent channel
selectivity performance
LO signal must have low spurious signals as they will cause receiver spurious
responses
must oscillate despite temperature and power supply variations
low susceptibility to microphonics where external mechanical or acoustic stress could
modulate the LO frequency or amplitude
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1. Radio Receiver Systems
Figure 1.1 Typical dual-conversion receiver.
1st IF filter:
protects its following stages from close-in IM signals
provides adjacent channel selectivity
attenuates the second image
equivalent noise bandwidth of the IF chain determines how much noise reaches the
detector and determines the modulation bandwidth that can be received
IF amplifier: high gain stage
raises the power level of the signal so that the baseband information can be recovered
easily
its intercept point must be high if it directly follows the mixer
if it follows one stage of IF filtering, the intercept point requirements can be relaxed
because the IF filter offers some protection against high-level, off-channel signals
The steps required for deciding the specifications for individual circuit blocks shown in
Fig.1.1 can be summarised as follows:
1. Allocate approximate gains and losses as needed to meet the required receiver
sensitivity specification and IM distortion requirements
2. Select the first IF frequency
3. Select the first LO injection side
4. Investigate the mixer
5. Based on mixer performance, design the injection filter and select LO technology
6. Investigate filter topologies
7. Design the RF amplifier
This type of receiver (Fig.1.1) is known as a superheterodyne receiver because it uses
frequency conversion, converting the relatively high RF carrier frequency to a lower IF
frequency before final demodulation.
Since large in-band interferers accompany the received signal even after the front-end
BPF, the nonlinearity of the following stages, particularly that of the low-noise amplifier
and the mixer becomes important. As illustrated in Fig.1.2, odd-order nonlinearities yield
intermodulation products that fall in the same desired channel. As third-order distortion is
usually dominant, the IP3 of each stage must be sufficiently high to avoid corruption of
the signal by the intermodulation products. Although distorting the amplitude, this effect
is important even if the signal carries information only in its phase or frequency, because
the zero-crossing points of the desired signal are corrupted by the intermodulation
product.
The BPF must exhibit minimal in-band loss while adequately suppressing the harmonics
and out-of-band spurious components of the transmitted signal.
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1. Radio Receiver Systems
Figure 1.2 Effect of nonlinearity in the front end.
The out-of-channel intermodulation products created by the power amplifier cannot be
suppress by the BPF and must be acceptably small by design.
Another important concern in the design of transceivers is the dynamics range of the
signals. With multipath fading and path loss, the required dynamics range for the received
signal is typically greater than 100dB. As the minimum detectable signal is in the
microvolts range, not only the input noise of the receiver but also cross-talk become
critical. (see Fig.1.3).
At the other extreme of the dynamic range, the receiver may experience large signals, for
example, amplitudes as high as several hundred millivolts, if it is close to a transmitter.
While the signal amplitude is not critical in PM and FM systems, the receive path must
still process the signal correctly. This issue leads to the use of automatic gain control
(AGC).
1.2.1 Antenna
The function of an antenna is to convert an RF signal from a transmitter to a propagating
electromagnetic wave or, conversely, convert a propagating wave to an RF signal in a
receiver. In a transceiver, where a transmitter and a receiver are co-located for full-duplex
communications, the same antenna may be used for both transmit and receive.
Some of the more obvious characteristics of an antenna include operating frequency
range, size, and pattern coverage.
The radiating pattern of an antenna is a plot of the transmitted or received signal strength
versus position around the antenna. Wireless systems that provide broadcast-type service,
such as television and AM/FM radio require antennas with pattern coverage that is
uniform in all directions. Such patterns are called omnidirectional, and can be obtained
with wire dipole and monopole antennas. Others systems such as point-to-point radio and
DBS receivers, require antennas that radiate (or receive) power preferentially in one
direction.
The measure of the directionality of an antenna pattern is provided by the directivity or
gain of the antenna; an omnidirectional antenna has low gain while a highly directive
antenna has high gain.
Because of the nature of the electromagnetic operation of an antenna, effective radiation
of a signal requires that the antenna have minimum physical dimensions on the order of
the electrical wavelength (λ=c/f) at the operating frequency. This means that antenna size
decreases with an increase in frequency, so that antennas at low frequencies will be very
large, while antennas at microwave frequencies and higher may be very small.
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1. Radio Receiver Systems
Figure 1.3 Desensitisation of LNA by PA output leakage.
In additional, it can be shown that the gain of an antenna is proportional to its crosssectional area divided by λ2, so that high antenna gain requires an electrically large
antenna.
More sophisticated antennas, such as phase array antennas, are able to change the
direction of their main beam electronically. Phase array antenna technology can be very
useful in commercial wireless systems because the antenna beam can be directed at a
given user while avoiding interference from other users. Such systems are called adaptive
arrays or smart antennas and may lead to increased channel capacity for cellular and PCS
telephone systems if cost reductions can be achieved.
1.2.2 Filters
Filters are two port components that are used to selectively pass or reject signals on the
basis of frequency.
An ideal low-pass filter (LPF) will pass all frequency components below its cutoff
frequency, while rejecting higher frequency components. Similarly, a high-pass filter
(HPF) will pass frequency components above its cutoff frequency, while rejecting lower
frequencies. A bandpass filter (BPF) passes frequency components within a narrow
passband, while rejecting frequency components outside the passband.
Filters are key components in all wireless transmitters and receivers. They are used to
reject interfering signals outside the operating band of receivers and transmitters, to reject
unwanted products from the outputs of mixers and amplifiers, and to set the IF bandwidth
of receivers.
Important filter parameters include the cutoff frequency, insertion loss and the out-ofband attenuation rate, measured in dB per decade of frequency. Filters with sharper cutoff
responses provide more rejection of out-of-band signals. Insertion loss, measured in dB, is
the amount of attenuation seen by signals through the passband of the filter.
Another important consideration is size and integrability with other circuit components.
At the present time, it is not possible to construct higher performance bandpass filters in
integrated circuits form. The inherent losses associated with RF and microwave integrated
circuits leads to filters having relatively high insertion losses and low out-of-band
attenuation rates.
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1. Radio Receiver Systems
For this reason, most wireless systems today use individual “off-chip” filters that are
located on the circuit board, rather than fully integrated filters. This results in a larger and
more costly assembly, but critical filtering performance is optimised.
1.2.3 Amplifiers
There are three main categories of amplifiers used in wireless system: low-noise
amplifiers (LNAs), used in the input stage of a receiver: power amplifiers (PAs), used in
the output stage of a transmitter; and IF amplifiers used in the IF stages of both receivers
and transmitters.
Important specifications for amplifiers include the power gain (in dB), the noise figure,
and the intercept points.
The noise figure of an amplifier is a measure of how much noise is added to the amplified
signal by the amplifier circuitry. This is most crucial in the front end of a receiver, where
the input signal level is very small, and it is desired to minimise the noise added by the
receiver circuitry. The noise power in a receiver is affected more by the first few
components than by later components. Thus it is imperative that the first amplifier in a
receiver has as low a noise figure as possible.
Because transistors are nonlinear devices, transistor amplifiers exhibit nonlinear effects.
Two important phenomena that occur in amplifiers because of these effects are saturation
and harmonic distortion. At low signal levels the output power of an amplifier is linearly
proportional to the input power. But because the output voltage of an amplifier cannot
exceed the bias voltage level, output power gradually reaches a saturation point as input
power increases. Saturation is usually only an issue with power amplifiers.
A more prevalent problem is related to the fact that harmonics of input signals are
generated at the output of an amplifier, and in the case of multiple input signal
frequencies, some of these harmonics will lie within the passband of the amplifier. These
harmonics lead to signal or harmonics distortion.
Generally the power level of these distortion harmonics is very low but the power level of
some of these distortion products increases as the cube of the input signal level. The
implication of this effect is that distortion power can be significant even for input power
levels well below the saturation point of an amplifier.
1.2.4 Mixers
A mixer is a three-port component that ideally forms the sum and difference frequencies
from two sinusoidal inputs. This allows the important function of frequency conversion to
be performed in superheterodyne transmitters and receivers.
In the case of a transmitter, the modulated baseband signal is up-converted in frequency
by mixing with a high frequency local oscillator signal. In a superheterodyne receiver, the
received signal is down-converted in frequency by mixing with a local oscillator to
produce a difference frequency (the IF frequency). In both cases, filters are required to
select the desired frequency products, while rejecting undesired frequencies that are
produced as a by-product of the mixing operation.
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1. Radio Receiver Systems
Modern mixers generally use diodes or transistors and produce many frequencies, based
on the harmonics of the input signals and their combinations, in addition to the desired
sum and difference frequencies.
A passive mixer (one that uses diodes) always produces an output signal (IF) of less
power than the input (RF) signal, because of dissipative losses in the mixer as well as
inherent losses in the frequency conversion process. This loss is characterised by the
mixer conversion loss. Mixers that use active components (e.g. transistors) generally have
lower conversion loss, and may even have conversion gain.
As in the case of amplifiers, harmonic distortion and noise are also important
considerations in mixer performance. Overall, mixer design usually involves trade-offs
between noise performance and conversion loss.
1.2.5 Oscillators
Oscillators are required in wireless receivers and transmitters to provide frequency
conversion, and to provide sinusoidal sources for modulation. Often these sources must
be tunable over a set of frequency range, and must provide very accurate output
frequencies.
The simplest oscillator uses a transistor with an LC network to control the frequency of
oscillation. Frequency can be tuned by adjusting the values of the LC network
electronically with a varactor diode. Such oscillators are simple and inexpensive, but
suffer from the fact that the output frequency is very susceptible to variations in supply
voltage, changing load impedances and temperature variations.
Better frequency control can be obtained by using a quartz crystal in place of the LC
resonator. A crystal-controlled oscillator (XCO) can provide a very accurate output
frequency, especially if the crystal is in a temperature controlled environment.
Crystal oscillators, however, cannot easily be tuned in frequency. A solution to this
problem is provided by the phase-locked loop (PLL), which uses a feedback control
circuit and an accurate reference source (usually a crystal-controlled oscillator) to provide
an output that is tunable with very high accuracy.
Phase-locked loops and other circuits that provide accurate and tunable frequency outputs
are called frequency synthesizers. Important parameters that characterise frequency
synthesizers are tuning range, frequency switching time, frequency resolution, cost and
power consumption.
1.2.6 Heterodyne Receivers
The most popular type of receiver used today is the superheterodyne circuit as shown in
Fig.1.4. The IF frequency is generally selected to be between the RF frequency and
baseband.
At microwave and millimetre wave frequencies, it is often necessary to use two stages of
down conversion to avoid problems due to LO stability. The dual-conversion
superheterodyne receiver of Fig.1.4 employs two local oscillators and mixers to achieve
down conversion to baseband with two IF frequencies.
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1. Radio Receiver Systems
Figure 1.4 Dual-IF heterodyne receiver.
Filtering a narrow channel that is centred at high frequencies and is accompanied by large
interferers demands prohibitively high Q’s. In heterodyne architectures, the signal band is
translated to much lower frequencies so as to relax the Q required of the channel-select
filter.
This translation is carried out by means of a mixer as shown in Fig.1.5a. To bring the
centre frequency from ω1 to ω2, the signal is first mixed with a sinusoid A0cosω0t, where
ω0=ω1−ω2, thereby yielding a band around ω2 and another around 2ω1−ω2. A low-pass
filter then removes the latter. This operation is called down conversion. Because of its
typically high noise, the down conversion mixer is preceded by a low-noise amplifier as
shown in Fig.1.5b.
For a heterodyne architecture with two input frequencies, x1(t)=A1cosω1t and
x2(t)=A2cosω2t, the bands symmetrically located above and below the LO frequency are
down-converted to the same centre frequency (Fig.1.6). If the received band of interest is
centred around ω1 (=ωLO−ωIF), then the image is around 2ωLO−ω1 (=ωLO+ωIF) and vice
versa.
This problem of image is a serious one. While each wireless standard imposes constraints
upon the signal emissions by its own users, it may have no control over the signals in
other bands. The image power can therefore be much higher than that of the desired
signal, requiring proper image rejection.
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1. Radio Receiver Systems
Figure 1.5 (a) Simple heterodyne down conversion, (b) inclusion of an LNA to lower the
noise figure.
Figure 1.6 Problem of image in heterodyne reception.
The most common approach to suppressing the image is through the use of an imagereject filter, place before the mixer. As depicted in Fig.1.7, the filter is designed to have a
relatively small loss in the desired band and a large attenuation in the image band. These
two requirements can be simultaneously met if 2ωIF is sufficiently large.
Fig.1.8 shows two cases corresponding to high and low values of IF to illustrate the tradeoff. A high IF leads to substantial rejection of the image whereas a low IF allows great
suppression of nearby interferers. The choice of IF therefore depends on trade-offs among
three parameters: the amount of image noise, the spacing between the desired band and
the image, and the loss of the image-reject filter.
Therefore, the heterodyne architecture exhibits a trade-off between image rejection and
channel selection. And since the image degrades the sensitivity of the receiver, the choice
of IF entails a trade-off between sensitivity and selectivity.
The selectivity and sensitivity of the heterodyne architecture have made it the dominant
choice in RF systems for many decades. Despite the complexity and the need for a large
number of external components, heterodyning is still viewed as the most reliable
reception technique.
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1. Radio Receiver Systems
Figure 1.7 Image rejection by means of a filter.
Figure 1.8 Rejection of image versus suppression of interferers for (a) high IF and (b) low IF.
Pros:
Superior sensitivity and selectivity performance
Cons:
Require high Q components
Low level of monolithic integration
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1. Radio Receiver Systems
1.2.7 Direct Conversion (Homodyne or Zero-IF) Receivers
The direct conversion uses a mixer and local oscillator to perform frequency down
conversion with a zero IF frequency.
The LO frequency is equal to the input carrier frequency as shown in Fig.1.9, which is
then converted directly to baseband. The channel selection requires only a low-pass filter
with relatively sharp cut-off characteristics.
The circuit of Fig.1.9(a) operates properly only with double sideband AM signals because
it overlaps positive and negative parts of the input spectrum.
For frequency and phase modulated signals, the down conversion must provide
quadrature outputs (see Fig.1.9(b)) so as to avoid loss of information, This is because the
two sides of FM or QPSK spectra carry different information and must be separated into
quadrature phases in translation to zero frequency.
The simplicity of the homodyne architecture offers two important advantages over a
heterodyne receiver. First, the problem of image is circumvented because ωIF=0. As a
result, no image filter is required. The sum frequency is twice the LO and easily filtered.
Second, the IF SAW filter and subsequent down conversion stages are replaced with lowpass filters and baseband amplifiers that are amenable to monolithic integration.
Direct conversion receivers are simpler and less costly than superheterodyne receivers
since there is no IF amplifier, IF bandpass filter or IF local oscillator required for the final
down conversion.
If the homodyne architecture is so simple, why has it not become popular in RF systems?
Direct translation of the spectrum to zero frequency entails a number of issues that do not
exist or are not serious in a heterodyne receiver. Examples are:
Channel selection – Rejection of out-of-channel interferers by an active low-pass filter is
more difficult than by a passive filter, fundamentally because active filters exhibit much
more severe noise-linearity-power trade-offs than do their passive counterparts.
DC offsets – Since in a homodyne topology the down-converted band extends to zero
frequency, extraneous offset voltages can corrupt the signal and more importantly,
saturate the following stages. Consider the receiver shown in Fig.1.10, where the LPF is
followed by an amplifier and an A/D converter.
First, the isolation between the LO port and the inputs of the mixer and LNA is not
infinite; that is, a finite amount of feedthrough exists between the LO port to points A
and B (Fig.1.10a). This is called LO leakage. The leakage signal appearing at the inputs
of the LNA and the mixer is now mixed with the LO signal, thus producing a DC
component at point C. This phenomenon is called self-mixing. A similar effect occurs if
a large interferer leaks from the LNA or mixer input to the LO port and is multiplied by
itself (Fig.1.10b).
Second, the total gain from the antenna to point X is typically around 80 to 100dB so as
to amplify the microvolt input signal to a level that can be digitised by a low cost, low
power ADC. Of this gain, typically 25 to 30dB is contributed by the LNA/mixer
combination.
In addition to introducing DC offsets, leakage of the LO signal to the antenna and
radiation therefrom creates interference in the band of other receivers using the same
wireless standard.
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1. Radio Receiver Systems
Figure 1.9 (a) Simple homodyne receiver, (b) homodyne receiver with quadrature down
conversion.
Figure 1.10 Self-mixing of (a) LO signal, (b) a strong interferer.
I/Q mismatch – For phase and frequency modulation schemes, a homodyne receiver
must incorporate quadrature mixing. This requires shifting either the RF signal or the LO
output by 90°. Shifting the RF signal generally entails severe noise-power-gain tradeoffs, making it more desirable to use the topology in Fig.1.11b. In either case, the errors
in the nominally 90° phase shift, and mismatches between the amplitudes of the I and Q
signals corrupt the down-converted signal constellation, thereby raising the bit error rate.
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1. Radio Receiver Systems
Figure 1.11 Quadrature generation in (a) RF path, (b) LO path.
Even-order distortion – Even order nonlinearity becomes problematic in homodyne down
conversion. This is illustrated in Fig.1.12 with two strong interferers close to the channel
of interest experience a nonlinearity such as y(t)=α1x(t)+α2x2(t) in the LNA. If
x(t)=A1cosω1t+A2cosω2t, then y(t) contains a term, α2A1A2cos(ω1−ω2)t, indicating that
two high frequency interferers generate a low frequency beat in the presence of evenorder distortion.
Flicker noise – Since the down-converted spectrum extends to zero frequency, the 1/f
noise of devices substantially corrupts the signals. This is a severe problem in MOS
implementation. For this reason, it is desirable to achieve a relatively high gain in the RF
range through the use of active mixers rather than passive mixers. As the stages
following the mixer operate at relatively low frequencies, they can incorporate very large
devices to minimise the magnitude of the flicker noise.
Pros:
Simpler and less costly than superheterodyne receivers
No image problem, thus no need for image reject filter
Use low pass filter, not band pass filter
Eliminate many off-chip components
High level of monolithic integration
Cons:
LO spurious leakage at antenna
DC offsets – LO self mixing
Strong interferers self mixing
Require offset cancellation
Flicker noise (1/f noise)
1.2.8 Digital-IF Receivers
Before the transformation from baseband to a RF channel, the waveform is digitised to
utilise the advantages of digital modulation. Coding is applied to the signal to more
efficiently use the available bandwidth and to minimise the effects of noise and
interference that will be introduced by the channel. The coded signal is filtered,
modulated and changed back to an analogue wave that is converted to the desired
frequency of transmission. Finally, the RF signal is filtered and amplified before it is
transmitted from the antenna.
At the receiver, the first IF signal is digitised, mixed with the quadrature phases of a
digital sinusoid, and low-pass filtered to yield the quadrature baseband signals as shown
in Fig.1.13. Digital processing avoids the problem of I and Q mismatch.
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1. Radio Receiver Systems
Figure 1.12 Effect of even-order distortion on interferers.
Figure 1.13 Digital-IF receiver.
After down conversion to the IF, the signal is separated into two distinct paths. To convert
to baseband, each path is mixed with an LO whose frequency equals the IF frequency.
The upper-path signal (I) is simply mixed with the LO and then filtered. In the lowerpath, a 90° phase shift is introduced in the mixing signal. This lower-path (Q) is converted
to baseband by mixing with the phase-shifted LO signal, and then filtered. This process
produces the I and Q baseband components of the data stream.
The principal issue in this approach is the performance required for the A/D converter:
- since the signal level at point A is typically no higher than a few hundred
microvolts, the quantization and thermal noise of the ADC must not exceed a few
tens of microvolts
- if the first IF bandpass filter cannot adequately suppress adjacent interferers, the
nonlinearity of the ADC must be sufficiently small to minimise corruption of the
signal by intermodulation
- the ADC dynamic range must be wide enough to accommodate variations in the
signal level due to path loss and multipath fading
- the ADC must achieve an input bandwidth commensurate with the value of IF while
consuming a reasonable amount of power
The ADC performance limitation can be partially alleviated as most ADCs incorporate
sample-and-hold circuits and hence can perform down conversion. The sampling IF
configuration is depicted in Fig.1.14 where the ADC samples the signal at a rate slightly
below fIF. The spectrum of the down-converted, digitised signal thus lies around fIF−fS.
This operation is followed by quadrature mixing and filtering to translate the spectrum to
the baseband. In this receiver, the IF signal is digitalised. The sampled data stream from
the ADC is digitally demodulated into its I and Q components and the original signal is
reconstructed.
Digital IF and sampling IF architectures have not been used in portable terminals because
of the ADC performance limitations.
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1. Radio Receiver Systems
Figure 1.14 Sampling IF architecture.
1.2.9 Subsampling Receivers
RF input can be sampled at a much lower rate because narrowband signals exhibit only a
small change from one carrier cycle to the next. The idea is that a bandpass signal with
bandwidth ∆f can be translated to a lower band if sampled at a rate equal to or greater
than 2∆f. This operation creates replicas of the spectrum with no aliasing as shown in
Fig.1.15.
Due to the large reduction in the down conversion rate, the use of subsampling can
simplify the design of the local oscillator and its associated synthesiser loop. Despite
these features, subsampling suffers from aliasing of noise.
Pros:
Digital implementation at IF
Digital programmable IF filter
Support multi-standard
Cons:
Large dynamic range of ADC
Aliasing of noise
Require high IP3 RF components
High power consumption
1.3 System Specifications
1.3.1 System Noise and Noise Floor
Assuming that a system’s gain is sufficient, the weakest signal that may be processed
satisfactorily, a figure of merit referred to as noise floor or minimum detectable signal
(MDS). For a given system noise power, the MDS determines the minimum signal-tonoise ratio (SNR) at the demodulator of the receiver.
The minimum SNR at the output of a receiver is sometimes expressed in terms of the
signal plus noise and distortion-to-noise and distortion ratio (SINAD). The SINAD is
related to SNR as
SINAD =
S+N
S
= 1+
N
N
Knowing the minimum SNR or SINAD and the noise characteristics of the receiving
system allows us to calculate the minimum detectable signal power.
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1. Radio Receiver Systems
Figure 1.15 Subsampling in (a) time and (b) frequency domains.
The noise power is equal to
Pn = kTB
where Pn is the noise power, k is the Boltzmann’s constant (1.38x10-23 watts per Kelvin),
T is the temperature in Kelvins, and B is the bandwidth (in Hertz) in which the noise
appears.
If the noise figure and bandwidth are known, the noise floor can be calculated using the
equation
Noise Floor = −174dBm + NF + 10 log B
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1. Radio Receiver Systems
1.3.2 Signal-to-Noise Ratio and Sensitivity
Successful radiocommunication depends on the achievement of a specified minimum
ratio of signal power to noise power, at the output of the receiver. The input voltage,
expressed in absolute units or decibels relative to a microvolt (dBµV) necessary to
achieve a particular signal-to-noise ratio in a particular bandwidth may be specified as a
figure of merit called sensitivity.
To calculate the sensitivity, we write
NF =
=
SNRin
SNRout
Psig / PRS
SNRout
where Psig denotes the input signal power and PRS the source resistance noise power, both
per unit bandwidth. Since the overall signal power is distributed across the channel
bandwidth, B, the two sides of the above equation must be integrated over the bandwidth
to obtain the total mean square power. Thus, for a flat channel,
Psig ,tot = PRS ⋅ NF ⋅ SNRout ⋅ B
The above equation predicts the sensitivity as the minimum input signal that yields a
given value for the output SNR. We have
Pin , min (dB ) = PRS (dBm / Hz ) + NF (dB ) + SNRmin (dB ) + 10 log B
= −174dBm + NF + 10 log B + SNR min
where Pin,min is the minimum input level that achieves SNRmin and B is expressed in Hertz.
Since Pin,min is a function of the bandwidth, a receiver may appear very sensitivity simply
because it employs a narrowband channel (at the cost of low information rate).
1.3.3 Noise Factor and Noise Figure
The noise floor predicted by Pn=kTB cannot be achieved and maintained in any real
network or system of networks because all real networks generate noise. Determining
how closely the SNR achieved at a given input level approaches the SNR achievable at
that input level in a noiseless network is therefore of high interest to the circuit and
system designer.
The degree to which a network’s noise contribution degrades the noise floor is evaluated
by its noise factor (F), which is expressed as the ratio
N + N added
F = in
N in
where Nin is the noise power available from the source, and Nadded is the noise power
added by the network, with both powers determine in the same bandwidth.
Expressing this ratio in dB returns the noise figure (NF=10logF), a bandwidth
independent figure of merit of great value in evaluating the noise performance of
networks and communication systems. It can be viewed as a measure of the degradation
in the signal-to-noise ratio between the input and output of the component.
We can also express NF as the ratio of the network’s input SNR to its output SNR,
S / N in
NF = in
S out / N out
17
1. Radio Receiver Systems
where NF is the noise figure in dB, S is the signal power, N is the noise power, and all
powers determined in the same bandwidth. Noise figure is a measure of how much the
SNR degrades as the signal passes through a system.
Noise figure can also be defined for antennas and antenna systems,
N / N ant
NFant = t
Nt
where NFant is the antenna system noise figure in dB, Nt is the antenna system’s thermal
noise power and Nant is the total noise power picked up by the antenna system.
1.3.4 Intermodulation
The new signals produced through intermodulation distortion (IMD) can profoundly
affect the performance even of systems operated far below gain compression.
IMD products of significant power can appear at frequencies remote from, in, and/or near
the system passband, resulting in demodulation errors (in reception) and interference to
other communications (in transmission).
Where an IMD product appears relative to the passband depends on the passband width
and centre frequency, the frequencies of the signals present at the system input, and the
order of the nonlinearity involved. These factors also determine the strength of an IMD
product relative to the desired signal.
If a weak signal accompanied by two strong interferers experiences third-order
nonlinearity, then one of the IM products falls in the band of interest, corrupting the
desired component, as shown in Fig.1.16.
To understanding how intermodulation arises, assuming,
x(t ) = A1 cos ω1t + A2 cos ω 2 t
For a system whose output does not depend on the past values of its input,
y (t ) = α 1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t )
= α 1 (A1 cos ω1t + A2 cos ω 2 t ) + α 2 (A1 cos ω1t + A2 cos ω 2 t ) + α 3 (A1 cos ω1t + A2 cos ω 2 t )
Expanding the left side and discarding dc terms and harmonics, we obtain the following
intermodulation products:
2
ω1 ± ω 2 = α 2 A1 A2 cos(ω1 + ω 2 )t + α 2 A1 A2 cos(ω1 − ω 2 )t
3α 3 A12 A2
3α A 2 A
cos(2ω1 + ω 2 )t + 3 1 2 cos(2ω1 − ω 2 )t
4
4
2
3α A A
3α A 2 A
2ω 2 ± ω1 = 3 2 1 cos(2ω 2 + ω1 )t + 3 2 1 cos(2ω 2 − ω1 )t
4
4
And these fundamental components
3
3
3
3


3
2
3
2
 α 1 A1 + α 3 A1 + α 3 A1 A2  cos ω1t +  α 1 A2 + α 3 A2 + α 3 A2 A1  cos ω 2 t
4
2
4
2




The components 2ω1−ω2 and 2ω2−ω1 appear in the vicinity of ω1 and ω2.
2ω1 ± ω 2 =
All the second-order products are undesired in an amplifier, but in a mixer the sum or
difference frequencies form the desired outputs. In either case, if ω1 and ω2 are close, all
the second-order products will be far from ω1 or ω2, and can easily be filtered (either
passed or rejected) from the output of the component.
18
3
1. Radio Receiver Systems
Figure 1.16 Corruption of a signal due to intermodulation between two interferers.
The corruption of signals due to third-order intermodulation of two nearby interferers is
so common and so critical that a performance metric, has been defined to characterise this
behaviour. Called the “third intercept point” (IP3), this parameter is measured by a twotone test in which A (=A1=A2) is chosen to be sufficiently small so that higher-order
nonlinear terms are negligible and the gain is relatively constant and equal to α1. As A
increases, the fundamentals increase in proportion to A, whereas the third-order IM
products increase in proportion to A3 (Fig.1.17a). Plotted on a logarithmic scale
(Fig.1.17b), the magnitude of the IM products grows at three times the rate at which the
main components increase. The third-order intercept point is defined to be at the
intersection of the two lines.
1.3.5 Dynamic Range
Dynamic range (DR) is generally defined as the ratio of the maximum input level that the
circuit can tolerate to the minimum input level at which the circuit provides a reasonable
signal quality.
Thermal noise sets the lower limit of the power span over which a network can operate.
The upper limit of a network’s power span is the level at which the power of one IM
product of a specified order is equal in power to the network’s noise floor.
The ratio of the noise floor power to the upper limit signal power is referred to as the
network’s dynamic range
(n − 1)(IPn,in − MDS )
DRn =
n
where DR is the dynamic range in dB, n is the order, IPin is the input intercept power in
dBm and MDS is the minimum detectable signal power in dBm.
The receiver dynamic range depends on the noise characteristics of the receiver as well as
the type of modulation being used, and the required minimum SNR.
1.3.6 Receiver Selectivity
Selectivity is a parameter that quantifies the tendency of a receiver to respond to channels
adjacent to the desired reception channel. As international regulations are gradually
moving to narrower channel spacings, receiver selectivity assumes greater importance
since it frequently limits system performance and places restrictions on frequency
allocation and system utilisation.
19
1. Radio Receiver Systems
Figure 1.17 Growth of output components in an intermodulation test.
[
]
Selectivity = −CR − 10 log 10 − IFsel / 10 + 10 − Spurs / 10 + BW ⋅ 10 SBN / 10
where
Selectivity = amount of adjacent channel selectivity relative to nominal receiver
sensitivity (dB)
CR = capture ratio, or cochannel rejection (dB)
IFsel = IF filter rejection at the adjacent channel (dB)
Spurs = LO spurious signals present in the IF bandwidth at a frequency offset equal to
the channel spacing (dBc)
BW = IF noise bandwidth (Hz)
SBN = SSB phase noise of LO at a frequency offset equal to the channel spacing
(dBc/Hz)
1.3.7 Receiver Spurious Responses
Receiver spurious responses are frequencies that are different from the desired receive
frequency, yet that still produce demodulated output in the receiver, if encountered at a
sufficiently high level. This situation is obviously undesirable and particularly
troublesome in receivers capable of tuning over a wide frequency range, because the RF
filters need to be wide to accommodate the wide frequency coverage.
Most receiver spurious responses are actually mixer spurious responses, which may or
may not be further attenuated by RF selectivity in the preceding stages. Most receiver
spurious responses result from harmonic mixing of the RF and LO signals. Any RF
frequency that satisfies the following relationship is a potential receiver spurious
response:
± mf RF ± nf LO = ± f IF
where
fRF = any incoming frequency into the mixer RF port
fLO = local oscillator frequency
fIF = desired IF frequency
m = integer multiplier of RF frequency
n = integer multiplier of LO frequency
1.3.8 Receiver Sensitivity
The minimum detectable signal power can be converted to a minimum detectable signal
voltage for a given receiver input impedance. This quantity is called the receiver
sensitivity. Assume receiver sensitivity is limited by thermal noise.
20
1. Radio Receiver Systems
Must consider device gains and noise figures, image noise, and local oscillator wideband
noise separately, and then combine them to produce an overall equivalent input noise
factor,
FT = FIN + F ' IN + F ' ' IN
FT = total equivalent input noise factor (linear)
FIN = total equivalent input noise factor derived from on-channel stage noise figures and
gains (linear)
F’IN = total equivalent input noise factor derived from image frequency stage noise
figures and gains (linear)
F’’IN = total equivalent input noise factor derived from local oscillator wideband noise
(linear)
The overall receiver sensitivity for a desired detector S/N ratio is given as
S
e = FT kTB RG
N
e = receiver sensitivity (V)
FT = total equivalent input noise factor (linear)
k = Boltzmann’s constant, 1.38x10−23 (J/K)
T = temperature (K)
B = equivalent noise bandwidth of system (Hz)
S/N = required S/N at detector output (linear)
RG = system impedance (Ω)
1.3.8.1 Contribution from Stage Gains and Noise Figures
In a typical wireless receiver, the input signal travels through a cascade of several
different components such as filters, amplifiers, mixers and transmission lines. Each of
these stages will progressively degrade the signal-to-noise ratio, so it is important to
quantify this effect to evaluate the overall performance of the receiver.
Noise figures and gains of all stages must be known to calculate the equivalent input
noise factor.
Friis formula for calculating cascaded noise figure is used to combine the stage
contributions and can be generalised for n stages,
n
F −1
FIN = 1 + ∑ i −1i
i =1
∏G j
j =0
FIN = equivalent input noise figure (linear)
Fi = stage noise factor (linear)
Gj = stage gain (linear, not in dB), G0=1
This relation proves especially useful if a receiver employs various off-the-shelf building
blocks that are characterised independently by manufacturers.
These results show that the noise characteristics of a cascaded system are dominated by
the first few stages, since the effect of the later stage is reduced by the product of the
gains of the preceding stages.
Thus for best overall system noise performance, the first stage of a receiver should have a
low noise figure and at least moderate gain. Expense and effort are most rewarded when
21
1. Radio Receiver Systems
applied to improving the noise characteristics of the first or second stage, as opposed to
later stages, since later stages have a diminished impact on overall noise performance.
1.3.8.2 Contribution from Image Noise
Image noise (Fig.1.18) is simply noise contained at the receiver’s image frequency which
is present at the first mixer’s RF port. This noise will be added to the down-converted
signal produced by an on-channel signal on a power basis. Thus, the analysis needs to be
carried up to the mixer stage that combines the two sidebands into one IF signal. Image
noise is down-converted to IF with similar conversion loss as the desired signal.
Must carefully sort out the different noise factors and gains that apply to the image
frequency and the on-channel frequency.
Carry out the normal analysis at the image frequency and normalise it by the ratio of the
overall image frequency gain to the on-channel frequency gain.
Filtering deliberately introduced to eliminate image noise by reflection mismatched case.
In the match case, the noise figure of a passive stage is equal to its loss. In the
mismatched case, we must allow zero noise figure to be assigned to a passive stage,
which provides image attenuation by reflection.
Image noise contribution to equivalent input noise factor is (assuming that the mixer
conversion loss is the same at the desired frequency as at the image frequency)
n


∏ G' j  n F ' −1 
j =1
i
1 +

F ' IN = n
i −1
 ∑

∏ G j  i =1 ∏ G' j 
j =1
j =0


F’IN = contribution to overall input noise factor by image noise (linear)
F’i = stage noise factor at image frequency (linear)
F = 1 for stage that provides image attenuation by reflection (ideally the stage
immediately preceding the mixer)
G’j = stage gain at image frequency (linear), G’0 = 1
Gj = stage gain on-channel (linear), G0 = 1
n = number of stages up to but excluding the mixer
1.3.8.3 Contribution from Wideband LO Noise
Wideband noise separated from the LO frequency and its harmonics by ±mfIF spacings
will mix to produce noise at the IF frequency. This is shown in Fig.1.18.
Noise at frequencies of ±fIF spacing from the LO harmonics also contributes and may be
dominant in some cases.
In contrast to image noise, wideband LO noise is down-converted to IF with a much
higher conversion loss than the desired signal.
The noise level in dBc/Hz has to be measured separately at nfLO+fIF and nfLO−fIF for each
n. This conversion of noise sidebands into IF noise is called mixer noise balance. It is
actually a conversion loss for the relevant sideband from the LO port to the IF port.
22
1. Radio Receiver Systems
Figure 1.18 Image noise and wideband LO noise contributions to overall noise figure
degradation.
There may be a bandpass injection filter between the LO and the mixer, its loss at the
appropriate noise sideband needs to be taken into account.
m
F ' ' IN = ∑
s =1
10 (PLO +Ws − Ls − M s ) / 10
n
1000kT0 ∏ G j
j =0
F’’IN = contribution to overall noise factor by wideband LO AM noise (linear)
PLO = local oscillator power (dBm)
Ws = wideband noise level of sideband s (dBc/Hz)
Ls = loss of injection filter at frequency of sideband s (dB)
Ms = mixer noise balance for sideband s (dB)
k = Boltzmann’s constant, 1.38x10−23 (J/K)
T0 = 290K or 16.85°C
Gj = gain of stage j (linear)
s = index for summation of noise powers at all sidebands of interest
m = number of sidebands taken into account
j = index to calculate gain up to and including the mixer
n = number of stages up to and including the mixer
The factor of 1000 comes from converting decibels relative to 1milliwatt to Watts.
1.3.9 Receiver Intercept Point
Intercept point is a measure of circuit or system linearity that allows us to calculate
distortion or IM product levels from the incoming signal amplitudes. The input intercept
point represents a fictitious input amplitude at which the desired signal components and
the undesired components (in this case the third-order products) are equal in amplitude as
shown in Fig.1.19.
The order of the intercept point refers to how fast the amplitudes of the distortion
products increase with an increase in input level. For example, for the third-order
intercept point (IP3), the IM product will increase in amplitude by 3dB when the input
signal is raised by 1dB.
23
1. Radio Receiver Systems
Figure 1.19 Third-order intercept diagram for a nonlinear component.
When a system is analysed for its overall equivalent intercept point resulting from the
combination of its constituent device intercept points, the most frequent assumption is
that the distortion products of the various stages are uncorrelated and add on a power
basis.
1.3.9.1 Second-Order Intercept Point
Second-order intercept point (IP2) is used to predict mixer performance with respect to a
particularly troublesome spurious response, called the half-IF (½IF) spurious response.
This receiver spurious response is separated from the desired channel by one-half IF
frequency.
The mechanism for ½IF generation is 2fRF±2fLO, where both harmonics are internally
generated, not fed in from the outside.
1
1
IF rejection = ( IP 2 − S − CR )
2
2
IP2 = equivalent input second-order intercept point at receiver input (dBm)
S = receiver sensitivity (dBm)
CR = capture ratio, or cochannel rejection (dB)
This analysis only needs to include the mixer intercept point because it is the stage that
generates the ½IF spurious response.
1.3.9.2 Third-Order Intercept Point
For small input powers, the third-order products must be very small, but will increase
quickly as input power increases.
24
1. Radio Receiver Systems
The output power of the first-order, or linear, product is proportional to the input power
and so the line describing this response has a slope of unity (before the onset of
compression) as shown in Fig.1.19. The line describing the response of the third-order
products has a slope of 3.
Both the linear and third-order responses will exhibit compression at high input powers,
the extensions of their idealised responses are shown with dotted lines. They will intersect
at a point above the onset of compression, as shown in the figure. This is the third-order
intercept point.
Third-order intercept point (IP3) is another important measure of system linearity. It is the
theoretical point at which the desired signal and third-order distortion products are equal
in amplitude. IP3 determines the amount of IM distortion produced in the receiver itself
when subjected to high-level interference.
The following procedure is used to calculate equivalent system input intercept point:
1. Draw block diagram of system with associated gains and IP3s.
2. Transfer all intercept points to system input, subtracting gains and adding losses
decibel for decibel.
3. Convert input intercept points to powers (dBm to milliwatts).
4. Assuming all intercept points are independent and uncorrelated, add powers “in
parallel”:
IPINPUT =
[mW]
1
1
1
+
+
IP1 IP2
1
+
IPN
5. Convert IPINPUT to dBm (milliwatts to dBm).
IP3 INPUT


1
= 10 log
 1
1
+
+

 IP1 IP2



1 
+

IPN 
IP3INPUT = equivalent system input intercept point (dBm)
IP1 = IP3 of first stage transferred to input (mW)
IPN = IP3 of last stage transferred to input (mW)
An example is shown in Fig.1.20.
IP3 INPUT




1


= 10 log
1
1
1
1 
1
+ +
+
 +

 ∞ 15.85 ∞ 19.95 100 
= 10 log(8.12) = 9.01 dBm
The equivalent system intercept point is 9.01dBm; the amplifier is the dominant
contributor because is has the lowest input equivalent intercept point.
For an ideal linear amplifier a plot of the output power versus input power is a straight
line with a slope of unity, and the gain of the amplifier is given by the ratio of the output
power to the input power. The amplifier response of Fig.1.19 tracks the ideal response
over a limited range, then begins to saturate resulting in reduced gain. This is the 1dB
compression point as the power level for which the output power has decreased by 1dB
from the ideal characteristics.
25
1. Radio Receiver Systems
Filter
Amplifier
Filter
G=-2dB
IP3=∞
G=12dB
IP3=10dBm
G=-3dB
IP3=∞
Mixer
G=-7dB
IP3=20dBm
Amplifier
G=22dB
IP3=20dBm
IP1 = ∞
IP2 = 10+2 = 12dBm=15.85mW
IP3 = ∞-12+2 = ∞
IP4 = 20+3-12+2 = 13dBm=19.95mW
IP5 = 20+7+3-12+2 = 20dBm=100mW
Figure 1.20 Example of system IP3 calculation.
The 1dB compression point (the level at which the gain drops by 1dB due to device
saturation) is about 11dB below the IP3 for amplifiers and 15dB below the IP3 for
mixers. The output power of an amplifier at the 1dB compression point will therefore be
about 11dB below the output IP3. The output intercept point equals the input intercept
point plus device gain.
Intermodulation distortion is a property of all systems that exhibit a nonlinear transfer
function, such as amplitude compression at sufficiently high levels. Third-order IM
distortion is most often produced when two signals, separated by ∆f and 2∆f from the
desired carrier beat together and produce on-channel interference.
Intermodulation rejection is the difference in decibels between sensitivity and input signal
level sufficient to produce interference can be calculated from the intercept point and
receiver sensitivity by use of
1
IM = (2 IP3 − 2 S − CR )
3
IM = intermodulation rejection (dB)
IP3 = equivalent input third-order intercept point (dBm)
S = receiver sensitivity (dBm)
CR = capture ratio, or cochannel rejection (dB)
26
1. Radio Receiver Systems
Example 1.1 Cascaded Noise Figure
st
st
Filter 1
Filter 2
1 mixer 1 IF filter
2nd IF filter
2nd IF amplifier
Detector
2nd mixer
1st IF amplifier
RF amplifier
Injection
filter
2nd LO
1st LO
Stage
Filter 1
RF amplifier
Filter 2
1st mixer
1st IF filter
1st IF amplifier
2nd IF filter
2nd mixer
2nd IF amplifier
NFtotal
Gain
(dB)
-0.5
12
-3
-7.5
-1.5
22
-5
-8
12
Noise figure
(dB)
0.5
2
3
7
1.5
3
5
8
20
Gain
(linear)
0.891
15.849
0.501
0.178
0.708
158.489
0.316
0.158
15.849
Noise figure
(linear)
1.122
1.585
1.995
5.012
1.413
1.995
3.162
6.310
100.000




n
F −1 
= 10 log 1 + ∑ i −1i
 i =1

Gj

∏


j =0

F −1
F −1
F −1
F −1
F −1 
F − 1 F3 − 1
F −1
= 10 log  F1 + 2
+
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9

G1
G1G 2 G1G 2 G3 G1 ...G 4 G1 ...G5 G1 ...G6 G1 ...G7 G1 ...G8 

= 10 log[1.122 + 0.657 + 0.07 + 0.567 + 0.328 + 1.116 + 0.015 + 0.119 + 14.032]
= 12.56dB or 18.026
Total gain = 20.5dB
Equivalent input noise figure = 18.02 or 12.56dB
Equivalent input noise temp = (NF−1)*290 = 4935.8K
27
1. Radio Receiver Systems
Example 1.2 Receiver Sensitivity Calculation
st
st
Filter 1
Filter 2
1 mixer 1 IF filter
2nd IF filter
2nd IF amplifier
Detector
2nd mixer
1st IF amplifier
RF amplifier
Injection
filter
2nd LO
1st LO
Stage
Filter 1
RF amplifier
Filter 2
1st mixer
1st IF filter
1st IF amplifier
2nd IF filter
2nd mixer
2nd IF amplifier
Detector
Gain
(dB)
-2.5
12
-2
-8
-1.5
20
-4
-8
20
Noise figure
(dB)
2.5
3.5
2
8.3
1.5
4
4
8
10
15
Gain
(linear)
0.562
15.849
0.631
0.158
0.708
100.000
0.398
0.158
100.000
Noise figure
(linear)
1.778
2.239
1.585
6.761
1.413
2.512
2.512
6.310
10.000
31.623
10dB
12kHz
-165dBc/Hz (flat with frequency)
23.5dBm
0dB at fLO±fIF offset
10dB at 2fLO±fIF
20dB at 3fLO±fIF
Mixer noise balance
30dB at fLO±fIF
25dB at 2fLO±fIF
20dB at 3fLO±fIF
6dB or 3.981
Required S/N at detector output
System impedance
50Ω
290K
Ambient temperature
Assume that the stage gains and the noise figures are the same at the image frequency as onchannel for all stages, except for filter 2, which has a 10dB loss and a noise figure of 0dB at
the image frequency. Neglect any contribution from the second image in the second IF.
Filter 2 image attenuation
Equivalent noise bandwidth
First LO wideband noise
First LO power
Injection filter attenuation
1. Total equivalent input noise factor derived from on-channel stage noise figures and gains
n
F −1
FIN = 1 + ∑ i −1i
i =1
∏G j
j =0
= 1.778 + 2.205 + 0.066 + 1.025 + 0.465 + 2.405 + 0.024 + 0.212 + 2.276 + 0.077
= 10.533
28
1. Radio Receiver Systems
2. Total equivalent input noise factor derived from image frequency stage noise figures and
gains before the mixer
The stage gains and the noise fiqures are the same at the image frequency as on-channel for
all stages, except for Filter 2 which has a 10dB loss and a noise figure of 0dB at the image
frequency.
Stage
Gain
Noise figure Gain
Noise figure
(dB)
(dB)
(linear)
(linear)
-2.5
2.5
Filter 1
0.562
1.778
12
3.5
RF amplifier
15.849
2.239
-10
0
Filter 2
0.100
1.000



F ' i −1 
j=

1+
= n
i −1
 ∑

i =1
G
G
'

∏1 j 
∏0 j 
j=
j=


n
F ' IN
∏1 G ' j 
n
0.562 * 15.849 * 0.1
[1.778 + 2.205 + 0]
0.562 * 15.849 * 0.631
= 0.631
=
3. Total equivalent input noise factor derived from local oscillator wideband noise up to and
including the first mixer
The LO wideband noise has 6 components, thus m=6.
m
F ' ' IN = ∑
s =1
10 (PLO +Ws − Ls − M s ) / 10
n
1000kT0 ∏ G j
[
j =0
= 10 (23.5−165−0 −30 ) / 10 + 10 (23.5−165− 0−30 ) / 10 + 10 (23.5−165−10− 25 ) / 10 + 10 (23.5−165−10 − 25 ) / 10 + 10 (23.5−165− 20− 20 ) / 10
][
]
+ 10 (23.5−165− 20 − 20 ) / 10 / 1000 * 1.38 *10 − 23 * 290 * (0.562 *15.849 * 0.631 * 0.158)
= 5.642
Total input equivalent noise factor,
FT = FIN + F ' IN + F ' ' IN
= 10.533 + 0.631 + 5.642
= 16.806
Receiver sensitivity,
e = FT kTB
S
RG
N
= 16.806 * 1.38 * 10 − 23 * 290 * 12000 * 3.981 * 50
= 0.401µV
Converting to dBm,
(
 0.401 * 10 −6
e = 10 log 
50

= −144.93dB
= −114.93dBm
) 
2

Note: the LO wideband noise contributes significantly to sensitivity degradation while
contributions from image noise may be neglected.
29