It is often desirable to translate a bandpass signal to a new center frequency. Frequency translation is used in the implementation of communications receivers as well as in a number of other applications. The process of frequency translation can be accomplished by multiplication of the bandpass signal by a periodic signal and is referred to asmixing. A block diagram of a mixer is given in Figure 3.18. As an example, the bandpass signal𝑚(𝑡) cos(2𝜋𝑓1𝑡)can be translated from𝑓1to a new carrier frequency𝑓2by multiplying it by a local oscillator signal of the form2 cos[2𝜋(𝑓1± 𝑓2)𝑡]. By using appropriate trigonometric identities, we can easily show that the result of the multiplication is
𝑒(𝑡) = 𝑚(𝑡) cos(2𝜋𝑓2𝑡) + 𝑚(𝑡) cos(4𝜋𝑓1± 2𝜋𝑓2)𝑡 (3.80) The undesired term is removed by filtering. The filter should have a bandwidth at least2𝑊 for the assumed DSB modulation, where𝑊 is the bandwidth of𝑚(𝑡).
A common problem with mixers results from the fact that two different input signals can be translated to the same frequency,𝑓2. For example, inputs of the form𝑘(𝑡) cos[2𝜋𝑓1± 2𝑓2)𝑡]
are also translated to𝑓2, since
2𝑘(𝑡) cos[2𝜋(𝑓1± 2𝑓2)𝑡] cos[2𝜋(𝑓1± 𝑓2)𝑡] = 𝑘(𝑡) cos(2𝜋𝑓2𝑡)
+𝑘(𝑡) cos[2𝜋(2𝑓1± 3𝑓2)𝑡] (3.81)
In (3.81), all three signs must be plus or all three signs must be minus. The input frequency 𝑓1± 2𝑓2, which results in an output at𝑓2, is referred to as theimage frequency of the desired frequency𝑓1.
To illustrate that image frequencies must be considered in receiver design, consider the superheterodyne receiver shown in Figure 3.19. The carrier frequency of the signal to be demodulated is𝑓𝑐, and the intermediate-frequency (IF) filter is a bandpass filter with center frequency𝑓IF, which is fixed. The superheterodyne receiver has good sensitivity (the ability to detect weak signals) and selectivity (the ability to separate closely spaced signals). This results because the IF filter, which provides most of the predetection filtering, need not be tunable.
Thus, it can be a rather complex filter. Tuning of the receiver is accomplished by varying the
Bandpass f ilter Center frequency
ω2
ω ω
ω m(t) cos 2t ω
m(t) cos 1t e(t)
Local oscillator
2 cos ( 1± 2)t
×
Figure 3.18 Mixer.
Radio- frequency (RF) f ilter
and amplif ier
Intermediate- frequency (IF) f ilter
and amplif ier
Demodulator Output
Local oscillator
×
Figure 3.19
Superheterodyne receiver.
frequency of the local oscillator. The superheterodyne receiver of Figure 3.19 is the mixer of Figure 3.18 with𝑓𝑐= 𝑓1and𝑓IF= 𝑓2. The mixer translates the input frequency𝑓𝑐 to the IF frequency𝑓IF.
As shown previously, the image frequency 𝑓𝑐± 2𝑓IF, where the sign depends on the choice of local oscillator frequency, also will appear at the IF output. This means that if we are attempting to receive a signal having carrier frequency𝑓𝑐, we can also receive a signal at𝑓𝑐+ 2𝑓IF if the local oscillator frequency is𝑓𝑐+ 𝑓IF or a signal at𝑓𝑐− 2𝑓IF if the local oscillator frequency is𝑓𝑐− 𝑓IF. There is only one image frequency, and it is always separated from the desired frequency by2𝑓IF. Figure 3.20 shows the desired signal and image signal for
f
f
f
f
f Desired
signal
Image signal Image signal at mixer output Local oscillator Signal at mixer output
1 + 2= LO
ƒ ƒ ƒ
1 + 2 ƒ ƒ 2
1 + 2
ƒ ƒ
2 3
1 + 2 = c + 2 lF
ƒ 2ƒ ƒ ƒ
2 = lF ƒ ƒ
2 = lF ƒ ƒ
1 = c ƒ ƒ
Passband of lF f ilter Figure 3.20
Illustration of image frequencies (high-side tuning).
Table 3.1 Low-Side and High-Side Tuning for AM Broadcast Band with𝒇𝐈𝐅=455 kHz Tuning range
Lower frequency Upper frequency of local oscillator
Standard AM 540 kHz 1600 kHz
broadcast band
Frequencies of 540 kHz -- 455 kHz 1600 kHz -- 455 kHz 13.47 to 1
local oscillator 85 kHz 1145 kHz
for low-side tuning
Frequencies of 540 kHz + 455 kHz 1600 kHz + 455 kHz 2.07 to 1
local oscillator = 995 kHz = 2055 kHz
for high-side tuning
a local oscillator having the frequency
𝑓LO= 𝑓𝑐+ 𝑓IF (3.82)
The image frequency can be eliminated by the radio-frequency (RF) filter. A standard IF frequency for AM radio is 455 kHz. Thus, the image frequency is separated from the desired signal by almost 1 MHz. This shows that the RF filter need not be narrowband. Furthermore, since the AM broadcast band occupies the frequency range 540 kHz to 1.6 MHz, it is apparent that a tunable RF filter is not required, provided that stations at the high end of the band are not located geographically near stations at the low end of the band. Some inexpensive receivers take advantage of this fact. Additionally, if the RF filter is made tunable, it need be tunable only over a narrow range of frequencies.
One decision to be made when designing a superheterodyne receiver is whether the frequency of the local oscillator is to be below the frequency of the input carrier (low-side tuning) or above the frequency of the input carrier (high-side tuning). A simple example based on the standard AM broadcast band illustrates one major consideration. The standard AM
2ƒlF
2ƒlF ƒLO
ƒLO
ƒ ƒ
ƒ
ƒ ƒ
c = LO + lF ƒi =ƒLO–ƒlF
ƒi =ƒLO +ƒlF ƒc =ƒLO–ƒlF
(a)
(b) Image signal
Image signal Desired signal
Desired signal
Figure 3.21
Relationship between𝑓𝑐 and𝑓𝑖(a) low-side tuning and (b) high-side tuning.
broadcast band extends from 540 kHz to 1600 kHz. For this example, let us choose a common intermediate frequency, 455 kHz. As shown in Table 3.1, for low-side tuning, the frequency of the local oscillator must be variable from 85 to 1600 kHz, which represents a frequency range in excess of 13 to 1. If high-side tuning is used, the frequency of the local oscillator must be variable from 995 to 2055 kHz, which represents a frequency range slightly in excess of 2 to 1. Oscillators whose frequency must vary over a large ratio are much more difficult to implement than are those whose frequency varies over a small ratio.
The relationship between the desired signal to be demodulated and the image signal is summarized in Figure 3.21 for low-side and high-side tuning. The desired signal to be demodulated has a carrier frequency of𝑓𝑐and the image signal has a carrier frequency of𝑓𝑖.