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Amplitude (Linear) Modulation Modulation is a process by which a parameter of a high frequency sinusoid is modified in accordance with the message signal to be transmitted. The high frequency sinusoid is known as the carrier and the message signal is the modulating signal. The modified carrier signal is called the modulated signal. A consequence of modulation is a translation or shifting of the message spectrum to a higher frequency band. Message signals, by nature, are low frequency or baseband signals. A baseband signal is a signal whose spectrum is positioned close to dc ( ω = 0 ). Examples of baseband signals include speech signals whose spectrum occupies the frequency band 0 to 3.5 kHz and video signals whose spectrum occupies the frequency band 0 to 4.3 kHz . There are two broad classes of communication – baseband communication and carrier communication. Modulation is required to match the signal to the channel (or link). Baseband communication requires no modulation whereas carrier communication requires modulation. Links such as local telephones using a pair of wires, coaxial cables and optical fibers do not need modulation. Radio links (radio and TV broadcast, microwave links, cellular phones and satellite links), on the other hand, must utilize modulation. There are two broad classes of modulation – linear (amplitude) modulation and nonlinear (angle) modulation. The reverse of modulation is called demodulation (or detection). Demodulation is a process which extracts the message signal from the modulated signal. In linear modulation the amplitude of the carrier signal is a linear function of the message signal. Depending on the nature of the spectral (frequency domain) relationship between the modulated signal and the message signal, we have the following types of linear modulation schemes: double-sideband suppressed carrier (DSB-SC) modulation, amplitude modulation (AM), single-sideband modulation (SSB), and vestigial-sideband modulation (VSB). Each of these schemes has its own distinct advantages, disadvantages, and practical applications. We will examine these different types of linear modulation schemes. The emphasis is characteristics such as signal spectrum, power and bandwidth, demodulation methods, and the complexity of transmitters and receivers. Need for Modulation Before we start a quantitative discussion and analysis of modulation systems, we need to examine the advantages of using modulated signals for information transmission. 1.
Ease of Radiation: If the communication channel consists of free space (radio channel), then antennas are needed to radiate and receive the signal. Efficient electromagnetic radiation requires antennas whose dimensions are the same order of magnitude as the wavelength ( λ = C / f - λ is the wavelength, C = 3x108 m / s is the speed of light and f is the frequency) of the signal being radiated. Many signals, including audio signals, have frequency components down to 100 Hz or lower. For these signals, antennas about 300 km long will be necessary if the signal is radiated directly. If modulation is used to impress the message signal on a high-frequency carrier, say at 100 MHz, then antennas need be no more than a meter or so across.
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2.
Multiplexing: If more than one signal utilizes a single channel, modulation may be used to translate different signals to different spectral locations thus enabling the receiver to select the desired signal. Applications of multiplexing include data telemetry, FM stereophonic broadcasting, and long distance telephone.
3.
Overcome Equipment Limitations: The performance of signal processing devices such as filters and amplifiers, and the ease with which these devices can be built depend on the signal location in the frequency domain and on the ratio of the highest to lowest signal frequencies. Modulation can be used for translating the signal to a location in the frequency domain where design requirements are easily met. Modulation can also be used to convert a “wideband signal” (a signal for which the ratio of highest to lowest signal frequencies is large) to a “narrow band” signal. Occasionally in signal processing applications, the frequency range of the signal to be processed and the frequency range of the processing apparatus may not match. If the processing apparatus is elaborate and complex, it may be wise to leave the processing equipment to operate in some fixed frequency range and, instead, translate the frequency range of the signal to correspond to this fixed frequency range of the equipment. Modulation can be used to accomplish this frequency translation.
5.
Frequency Assignment: Modulation allows several radio or television stations to broadcast simultaneously at different carrier frequencies and allows different receivers to be “tuned” to select different stations.
6.
Reduce Noise and Interference: The effect of noise and interference cannot be completely eliminated in a communication system. However, it is possible to minimize their effects by using certain types of modulation schemes. These schemes generally require a transmission bandwidth much larger than the bandwidth of the message signal. Thus bandwidth is traded for noise reduction - an important aspect of communication system design.
Double Sideband Suppressed Carrier Modulation In amplitude modulation the amplitude of a high-frequency carrier is varied in direct proportion to the low-frequency (baseband) message signal. The carrier is usually a sinusoidal waveform, that is, c(t ) = Ac cos(ωc t + θ c ) or c(t ) = Ac sin(ωc t + θ c ) Ac is the unmodulated carrier amplitude ωc = 2π f c is the unmodulated carrier angular frequency in radians/s; f c is the carrier frequency in Hz θ c is the unmodulated carrier phase, which we shall assume is zero. The amplitude modulated carrier has the mathematical form
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ϕ DSB − SC (t ) = A(t ) cos (ωc t ) A(t ) is the instantaneous amplitude of the modulated carrier, and is a linear function of the message signal m(t ) . A(t ) is also known as the envelope of the modulated signal. For double-sideband suppressed carrier (DSB-SC) modulation the amplitude is related to the message as follows: A(t ) = Ac m(t ) Consider a message signal with spectrum (Fourier transform) M (ω ) which is band limited to 2π B as shown in Figure 1(b). The bandwidth of this signal is B Hz and ωc is chosen such that ωc >> 2π B . Applying the modulation theorem, the modulated Fourier transform is 1 A(t ) cos(ωc t ) = m(t ) cos(ωc t ) ⇔ [ M (ω − ωc ) + M (ω + ωc ) ] 2 The time domain waveform and the frequency spectrum are shown in Figure 1(c). The dashed lines represent the positive ( + A(t ) = + m(t ) ) and negative ( − A(t ) = −m(t ) ) amplitudes, respectively. Properties of DSB-SC Modulation:
(a) (b) (c)
(d) (e)
(f)
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There is a 180o phase reversal at the point where + A(t ) = + m(t ) goes negative. This is typical of DSB-SC modulation. The bandwidth of the DSB-SC signal is double that of the message signal, that is, BWDSB − SC = 2 B (Hz) . The modulated signal is centred at the carrier frequency ωc with two identical sidebands (double-sideband) – the lower sideband (LSB) and the upper sideband (USB). Being identical, they both convey the same message component. The spectrum contains no isolated carrier. Thus the name suppressed carrier. The 180o phase reversal causes the positive (or negative) side of the envelope to have a shape different from that of the message signal, see Figure 1(d) and (e). This is known as envelope distortion, which is typical of DSB-SC modulation. The power in the modulated signal is contained in all four sidebands.
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(d) Modulating signal
(e) Modulated signal Figure 1 DSB-SC modulation. Generation of DSB-SC Signals
The circuits for generating modulated signals are known as modulators. We shall discuss three basic modulators – nonlinear, switching and ring modulators. Conceptually, the simplest modulator is the product or multiplier modulator, Figure 1(a). However, it is very difficult (and expensive) in practice to design a product modulator that maintains amplitude linearity at high carrier frequencies.
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Nonlinear Modulators
Nonlinear elements produce distortion by generating new signal components at new frequencies. We can utilize the distortions to generate modulated signals by filtering out all the undesired components. Consider two nonlinear devices (NL), in Figure 2, with identical input-output characteristics given by y (t ) = ax(t ) + bx 2 (t ) The input signal to the top device is the sum of the message and carrier signals x1 (t ) = cos(ωc t ) + m(t ) The input signal to the bottom device is the difference of the message and carrier signals x2 (t ) = cos(ωc t ) − m(t ) The output signals are given, respectively, by y1 (t ) = ax1 (t ) + bx12 (t ) = a ( cos(ωc t ) + m(t ) ) + b ( cos(ωc t ) + m(t ) )
2
= a ( cos(ωc t ) + m(t ) ) + b cos 2 (ωc t ) + 2bm(t ) cos(ωct ) + m 2 (t ) y2 (t ) = ax2 (t ) + bx22 (t ) = a ( cos(ωct ) − m(t ) ) + b ( cos(ωc t ) − m(t ) )
2
= a ( cos(ωct ) − m(t ) ) + b cos 2 (ωc t ) − 2bm(t ) cos(ωc t ) + m 2 (t )
Individually, these outputs contains several unwanted components in addition to the desired component 2bm(t ) cos(ωc t ) . Subtraction of the two outputs gives z (t ) = 2am(t ) + 4bm(t ) cos(ωc t ) The first component is the baseband message signal, whose spectrum is located around dc with a maximum frequency of ω = 2π B . The second component is the desired DSB-SC Since signal whose spectrum is centred at ωc with minimum frequency ωc − 2π B . ωc >> 2π B , the two spectra will not overlap. Therefore, passing z (t ) through a band-pass filter centred at ωc with bandwidth B , suppresses the baseband component and passes the desired modulated signal unaltered, that is, ϕ DSB − SC (t ) = 4bm(t ) cos(ωct )
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Without the band-pass filter, the circuit in Figure 2 is known as a single balanced modulator in the sense that it does not remove all the unwanted components. It needs a second stage, the band-pass filter, to remove the rest. The two together is called a double-balanced modulator. Semiconductor diodes are often used for nonlinear devices in balanced modulators. An example is shown in Figure 3. The performance of the modulator is dependent on how close the characteristics of the two diodes can be matched. Performance degrades if the parameters a and b are not identical for both diodes.
Figure 2
Figure 3
A nonlinear modulator for DSB-SC.
A circuit diagram of a double-balanced modulator.
Switching Modulator
Figure 4
A basic switching modulator.
A linear device with time-varying gain is used to obtain multiplier action. This is demonstrated in Fig. 4, in which the time-varying device is an analog switch that is turned on an off by a square-wave oscillator signal operating at a frequency of ωc . The switch is turned on for half
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the period ( Tc = 2π / ωc ) and off for the rest of the period of the pulse generator output. When the switch is on, the input is passed and when the switch is off the input is blocked. This way, we think of the gain of the switch as being either unity or zero. The waveform at the output of the analog switch is equivalent to the product of the input signal uin (t ) = m(t ) and the oscillator output s (t ) = w(t ) , that is, v1 (t ) = m(t ) w(t ) The signal v1 (t ) = m(t ) w(t ) , is shown in Figure 5(b).
Figure 5
Switching modulator for DSB-SC.
Since w(t ) is periodic, we can replace it by its Fourier series given by w(t ) =
∞ 1 sin(nπ / 2) cos(nωc t ) + 2∑ 2 nπ n =1
Therefore, ∞ sin(nπ / 2) ⎡1 ⎤ v1 (t ) = m(t ) ⎢ + 2∑ cos(nωc t ) ⎥ nπ n =1 ⎣2 ⎦ ∞ 1 2 sin(nπ / 2) = m(t ) + m(t ) cos(ωc t ) + 2∑ cos(nωc t ) 2 nπ π n=2
The desired term
2
π
m(t ) cos(ωc t ) , is obtained from the n = 1 term. This term is centred at ωc
while the terms in the summation are centred at ω = 2ωc , 3ωc ,... , which are very far away from
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1 m(t ) is baseband. Therefore, a band-pass filter centred at ωc with 2 2 bandwidth twice that of m(t ) , will pass m(t ) cos(ωc t ) and eliminate the rest. The modulated
ωc . Also, the first term
π
output is therefore given by
ϕ DSB − SC (t ) = km(t ) cos(ωc t ) Ring Modulator
The schematic diagrams and waveforms for a ring modulator are shown in Figures 6 and 7. Semi- conductor diodes are ideally suited for use in balanced modulator circuits because they are stable, require no external power source, have a long life, and require virtually no maintenance. A balanced modulator has two inputs: a single-frequency carrier and the modulating signal. For the modulator to operate properly, the amplitude of the carrier must be sufficiently greater than the amplitude of the modulating signal (approximately six to seven times greater). This ensures that the carrier and not the modulating signal controls the on or off condition of the four diode switches ( D1 to D4 ). Circuit operation: Diodes D1 to D4 are electronic switches that control whether the modulating
signal m(t ) is passed from input transformer T1 to output transformer T2 as is or with a 1800 phase shift. With the carrier polarity as shown in Figure 6 (b), diode switches D1 and D2 are forward biased and ‘on’, while diode switches D3 and D4 are reverse biased and ‘off’. Consequently, the modulating signal is transferred across the closed switches to T2 without a phase reversal. When the polarity of the carrier reverses, as shown in Figure 6(c), diode switches D1 and D2 are reverse biased and ‘off’, while diode switches D3 and D4 are forward biased and ‘on’. Consequently, the modulating signal undergoes a 1800 phase reversal before reaching T2 Carrier current flows from its source to the center taps of T1 and T2 where it splits and goes in opposite directions through the upper and lower halves of the transformers. Thus, their magnetic fields cancel in the secondary windings of the transformer and the carrier is suppressed. If the diodes are not perfectly matched or if the transformers are not exactly center tapped, the circuit is out of balance and the carrier is not totally suppressed. In practice, it is virtually impossible to achieve perfect balance; thus, a small carrier component is always present in the output signal. This carrier leak is usually considered negligible. Figure 7 shows the input and output waveforms associated with a balanced modulator for a single-frequency modulating signal. It can be seen that D1 and D2 conduct only during the positive half-cycles of the carrier input signal, and D3 and D4 conduct only during the negative half-cycles.
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(a)
(b)
Figure 6
(c) Operation of the ring modulator
The output from a balanced modulator consists of a series of pulses, which is equivalent to the product of the modulating signal m(t ) in Figure 6(a) and a rectangular pulse train whose frequency (or period) is determined by the carrier frequency Figure 6(b). Consequently, the output waveform takes the shape of the modulating signal Figure 6(c), except with alternating positive and negative polarities that correspond to the polarity of the carrier signal.
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Figure 7
Ring modulator waveforms
As before, this product is given by ∞ sin(nπ / 2) ⎡1 ⎤ v1 (t ) = m(t ) ⎢ + 2∑ cos(nωc t ) ⎥ nπ n =1 ⎣2 ⎦ ∞ 1 2 sin(nπ / 2) = m(t ) + m(t ) cos(ωc t ) + 2∑ cos(nωc t ) 2 nπ π n=2
Therefore, a band-pass filter (see Figure 8) centred at ωc with bandwidth twice that of m(t ) , will 2 pass the desired component m(t ) cos(ωc t ) and eliminate the rest. The modulated output is
π
therefore given by
ϕ DSB − SC (t ) = km(t ) cos(ωc t )
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Figure 8
A complete ring modulator
Demodulation of DSB-SC Signals
Demodulation or detection is the process of recovering the message signal from the modulated waveform. The method used to recover message signals from DSB-SC waveforms is known as coherent or synchronous detection (or demodulation). Coherent detection
The block diagram for coherent detection is shown in Figure 9. This is similar to the modulator except that the band-pass filter is replaced by a low-pass filter. The received DSB-SC signal is sm (t ) = ϕ DSB − SC (t ) = Ac m(t ) cos(ωc t ) The receiver first generates an exact (coherent) replica (same phase and frequency) of the unmodulated carrier sc (t ) = cos(ωc t ) The coherent carrier is then multiplied with the received signal to give
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sm (t ) sc (t ) = Ac m(t ) cos(ωc t ) cos(ωc t ) =
1 1 Ac m(t ) + Ac m(t ) cos(2ωc t ) 2 2
The first term is the desired baseband signal while the second is a band-pass signal centred at 2ωc . A low-pass filter with bandwidth equal to that of the m(t ) will pass the first term and reject the band-pass component.
Figure 9
Coherent demodulator for DSB-SC signals, s (t ) = Ac m(t )
The problem with this is that the local carrier must have exactly the same phase (phase coherent) as the incoming carrier. The additional circuit (carrier recovery circuit) that ensures phase coherence makes the receiver more complex and expensive. For example, suppose the there is a phase error θ . Then the output of the multiplier is sm (t ) sc (t ) = Ac m(t ) cos(ωc t ) cos(ωct + θ ) =
1 1 Ac m(t ) cos(θ ) + Ac m(t ) cos(2ωct + θ ) 2 2
The low-pass filter output is
1 1 s (t ) = Ac m(t ) cos(θ ) 2 2 There is no problem if the phase error is very small because cos(θ ) ≈ 1 . However, if the phase error is close to π / 2 , then cos(θ ) ≈ 0 and the signal is lost completely. Amplitude Modulation (AM)
Suppose we are given a message signal x(t ) . We first normalize this signal with respect to its maximum peak value X p = x(t ) peak , that is,
m(t ) =
x(t ) x(t ) = ⇒ −1 ≤ m(t ) ≤ 1 x(t ) peak Xp
We may now express x(t ) in terms of the normalized signal as x(t)=X p m(t )
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An AM signal is a DSB-SC signal plus a carries with a large amplitude, such that Ac ≥ X p ). That is, the AM signal is given by
ϕ AM (t ) = Ac cos(ωc t ) + x(t ) cos(ωc t ) = [ Ac + x(t ) ] cos(ωct )
⎡ Xp ⎤ = Ac ⎢1 + m(t ) ⎥ cos(ωc t ) Ac ⎣ ⎦ = Ac [1 + µ m(t )] cos(ωc t )
The parameter µ is called the modulation index or depth of modulation and takes on the range of values
0 < µ ≤1 Examples of time waveforms of AM signal are shown in Figure 10 for µ = 0.0, 0.3, 0.8 and 1.0 , respectively.
(a) µ = 0
(c) µ = 0.8
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(b) µ = 0.3
(d) µ = 1 Figure 10 AM envelopes for various values of µ
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Any value of the modulation index µ that is greater than 1 is referred to as over modulation. Over modulation causes zero crossing of the envelope and causes envelope distortion. Spectrum of AM Signals
Suppose the spectrum of the normalized message is M (ω ) ⇔ m(t ) . The spectrum of an AM signal, obtained by applying the modulation theorem, is given by 1 2
[π Acδ (ω − ωc ) + δ (ω + ωc )] + [ M (ω − ωc ) + M (ω + ωc )] ⇔ ϕ AM (t ) = Ac [1 + µ m(t )] cos(ωct ) An example of the spectrum of an AM signal is shown in Figure 11.
(a)
Figure 11
(b) Amplitude modulation, (a) Message spectrum and (b) AM spectrum.
The spectrum contains a discrete carrier plus sidebands. Since the carrier contains no message, power carried in this component is wasted. Power and Efficiency of an AM Signal
Recall the expression for the AM signal
ϕ AM (t ) = Ac [1 + µ m(t ) ] cos(ωc t ) = Ac cos(ωc t ) + µ Ac m(t ) cos(ωct )
The power in the carrier component is Pc =
Ac2 2
The power in the message (sidebands) term is obtained by squaring and integrating
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1 2 2 2 1 µ Ac m (t ) + µ 2 Ac2 m2 (t ) cos(2ωc t ) 2 2 1 = µ 2 Ac2 Pm 2
PSB =
The symbol m 2 (t ) = lim T →∞
the time averaged integral, 1 T /2 2 m (t )dt = Pm T ∫−T / 2
The second integral is equal to zero m 2 (t ) cos(2ωc t ) = lim T →∞
1 T
∫
T /2
−T / 2
m 2 (t ) cos(2ωc t )dt = 0
Therefore, the power in an AM signal is Ac2 1 2 2 + µ Ac Pm 2 2 = Pc + µ 2 Pm Pc
PAM =
= Pc + PSB
The efficiency is defined as
η= =
PSB Useful power = Total power PSB + Pc
µ 2 Pm 1 + µ 2 Pm
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Demodulation of AM Signals: Envelope Detector:
An envelope detector is a circuit whose output closely approximates the envelope of the input signal. The baseband (message) signal m(t ) can be recovered from the received AM signal xr (t ) = ϕ AM (t ) using the simple circuit shown in Figure 12 as long as −1 ≤ m(t ) ≤ 1 . If −1 ≤ m(t ) ≤ 1 , then the envelope of the received signal never crosses zero and so the positive portion of the envelope approximates the message signal. The positive envelope is recovered by rectifying xr (t ) = ϕ AM (t ) and smoothing the rectified waveform using an RC network or a lowpass filter.
(a) Envelope detector
(b) Correct demodulation
(c) RC too large
Figure 12
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(d) RC too small Envelope demodulation of AM signals.
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Since the diode only allows current to flow (conducts) in one direction, the negative portion is chopped off at the output of the diode, as shown in Figure 12. The operation of the envelope detector can be summarized as follows: (1) As the input waveform rises to its peak value, the diode conducts and current flows through the capacitor C . The capacitor charges up to the peak of the input. (2) As the input waveform starts to drop from its peak value, the diode is cut off and the capacitor discharges through the resistor R . (3) The charge and discharge operation is repeated throughout all the cycles of the input waveform. (4) The discharge rate is determined by the time constant RC . If RC is too small, the capacitor will discharge faster and not follow the envelope as shown in Figure 12 (d). (5) If RC is too large, the capacitor will not discharge enough to follow the envelope as shown in Figure 12 (c). (6) For best operation, the carrier frequency ωc should be much higher than the maximum frequency 2π B of the message signal and the discharge time constant RC should be adjusted so that 1 1 << RC << 2π B ωc In this case the capacitor will discharge just enough to closely follow the envelope of the input.
The output of the envelope detector consists of ripples, which is at the carrier frequency. The ripples may be removed by smoothing the detector output. Smoothing is accomplished by lowpass filtering. The output, after smoothing is given by yd (t ) = Ac + µ Ac m(t ) The dc component Ac is removed by a dc blocker, which is essentially a capacitor, which is placed in series. The AM signal may also be demodulated to recover the message signal with coherent detection followed by a dc blocker.
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Quadrature amplitude modulation (QAM)
QAM is a modulation method for transmitting two independent messages in an overlapping spectrum. This is accomplished by modulating one message on the cosine (inphase) carrier and the other message on the sine (quadrature) carrier. That is,
ϕQAM (t ) = Ac m1 (t ) cos(ωc t ) + Ac m2 (t )sin(ωc t ) Recovery of the individual message signals is done by coherent detection. Both the modulator and demodulator for QAM are shown in Figure 13.
Figure 13
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Quadrature amplitude modulation and demodulation.
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Single-Sideband Modulation (SSB)
In DSB-SC and AM, there are two sets of sidebands each of which convey identical message information. As a result, the bandwidth required for transmission is doubled that of the original signal. As it turns out, we can suppress one set of sidebands before transmission and accurately recover the original message signal at the receiver. We may, therefore, be able to transmit a completely independent message signal in the other sidebands. In short SSB modulations allows us to transmit two independent messages in the same bandwidth where AM and DSB-SC modulations would transmit only one message signal. Suppose the message bandwidth is B Hz . Then, the bandwidth required to transmit one message signal is also B Hz for SSB modulation. Single-sideband modulated signals can be generated by one two methods – the filtering method and the phase-shift method. (a)
Filtering Method
In the filtering method we first generate a DSB-SC signal and then pass it through a band-pass filter with centre frequency and bandwidth just enough to pass either the upper sideband or the lower sideband as illustrated in Figure 14. The filter is referred to as a sideband filter and its transfer function is denoted by H SSB ( f ) .
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Figure 14
Filtering method for generating an SSB (USB) signal
Because an ideal filter is not practically realizable, the filtering method can only be used when there is insignificant content around dc, that is, if there is a gap between the sideband as shown in the figure below. An example of a message signal with insignificant contents around dc is speech. On the other hand, data and video signals have significant dc contents (as shown in the figure below) and so there is no gap between the sidebands. The filter must be ideal, which is not realizable. (b) Phase-Shift Method
Fundamental to the phase-shift method is the Hilbert transform. We shall now provide a brief review of the Hilbert transform. Hilbert Transform:
A Hilbert transformer is a filter with the following transfer function:
H (ω ) = − j sgn(ω )
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The signum function sgn(.) is defined as ⎧1 ω >0 sgn(ω ) = ⎨ ⎩−1 ω < 0 Using this definition, we can express the transfer function in polar (or magnitude and phase) form as follows: ⎧ − j = e − jπ / 2 ω > 0 H (ω ) = ⎨ jπ / 2 ω<0 ⎩ j=e
Magnitude and phase response:
From the polar form expression, we see that the magnitude response is constant and equal to 1 , that is, H (ω ) = 1
Also, the phase response is given by ⎧ π ⎪⎪− 2 θ (ω ) = ⎨ ⎪π ⎪⎩ 2
ω >0 ω<0
This shows that the Hilbert transformer adds a phase shift of −π / 2 to all the positive frequencies of the input signal while the magnitude is unaltered. Impulse Response:
From the Fourier transform table, we find impulse response of the Hilbert transform to be given by h(t ) =
1 ⇔ H (ω ) πt
The Hilbert transform of a signal g (t ) , denoted by gˆ (t ) or g h (t ) is the output of the Hilbert transformer in response to the signal g (t ) , applied to the input. That is,
gˆ (t ) = h(t ) ∗ g (t ) ⇔ Gˆ (ω ) = (− j ) sgn(ω )G (ω )
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22 If g (t ) = Ac cos(ωc t ) , then gˆ (t ) = Ac cos(ωc t − π / 2) = Ac sin(ωct )
In the frequency domain G (ω ) = π [δ (ω − ωc ) + δ (ω + ωc ) ] Gˆ (ω ) = H (ω )G (ω )
= (− j ) sgn(ω )G (ω ) = π ⎡⎣δ (ω − ωc )e− jπ / 2 + δ (ω + ωc )e jπ / 2 ⎤⎦ Taking the inverse Fourier transform gives gˆ (t ) = Ac cos(ωc t − π / 2) = Ac sin(ωct ) . Example 2:
Let g (t ) = Ac m(t ) cos(ωc ) , then gˆ (t ) = Ac mˆ (t ) sin(ωc t )
Exercise 1:
(a) Show that Gˆ (ω ) = G (ω ) and therefore, 1 2π
2 ˆ (ω ) d ω = 1 G ∫−∞ 2π ∞
∫
∞
−∞
2
G (ω ) d ω = Eg
That is, the signal and its Hilbert transform contain the same energy (b)
∫
∞
−∞
g (t ) gˆ (t )dt = 0
That is, the signal and its Hilbert transform are orthogonal. Analytic Functions:
An analytic function is defined as
x(t ) = g (t ) + jgˆ (t ) Taking the Fourier transform gives X (ω ) = G (ω ) + j (− j ) sgn(ω )G (ω ) = G (ω ) [1 + sgn(ω )] Using the definition of the signum function we obtain ⎧2G (ω ) ω > 0 X (ω ) = ⎨ ω<0 ⎩ 0 We see that an analytic signal has positive spectrum.
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Phase-Shift Method for Generating SSB signals:
The SSB signal is obtained by adding or subtracting a DSB-SC signal and its Hilbert transform as follows:
ϕ SSB (t ) = ϕ DSB − SC (t ) ± ϕˆ DSB − SC (t ) = Ac m(t ) cos(ωc t ) ± Ac mh (t ) sin(ωct ) The minus sign gives upper sideband while the plus sign gives lower sidebands. A block diagram implementation of the above equation is shown in Figure 15.
Figure 15
Phase-shift method for generating SSB signals
Taking the Fourier transform of the SSB signal gives
ϕ SSB (ω ) =
Ac A [ M (ω − ωc ) + M (ω + ωc )] ± c [ M h (ω − ωc ) + M h (ω + ωc )] 2 2j
where M h (ω − ωc ) = (− j ) sgn(ω − ωc ) M (ω − ωc ) M h (ω + ωc ) = (− j ) sgn(ω + ωc ) M (ω + ωc ) Consider the upper sideband SSB, that is, pick the minus sign. Then we have
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ϕ SSB (ω ) =
24 Ac A [ M (ω − ωc ) + M (ω + ωc )] − c [ M h (ω − ωc ) − M h (ω + ωc )] 2 2j
⎡ ⎤ Ac ⎡ ⎤ 1 1 ⎢ M (ω − ωc ) − j (− j ) sgn(ω − ωc ) M (ω − ωc ) ⎥ + 2 ⎢ M (ω + ωc ) + j (− j ) sgn(ω + ωc ) M (ω + ωc ) ⎥ ⎣ ⎦ ⎣ ⎦ A A = c M (ω − ωc ) [1 + sgn(ω − ωc ) ] + c M (ω + ωc ) [1 − sgn(ω + ωc ) ] 2 2 =
Ac 2
By definition we have ⎧ 1 ω > ωc ⎧ 1 ω > −ωc and sgn(ω + ωc ) = ⎨ sgn(ω − ωc ) = ⎨ ⎩−1 ω < −ωc ⎩−1 ω < ωc These two functions are sketched below.
Therefore, ⎧ 2 ω > ωc ⎧0 ω > −ωc and 1 − sgn(ω + ωc ) = ⎨ 1 + sgn(ω − ωc ) = ⎨ ⎩2 ω < −ωc ⎩ 0 ω < ωc
These functions are also sketched below. Using these results, the Fourier transform of the SSB signal reduces to (also sketched in the figure below)
ω > ωc ⎧ Ac M (ω − ωc ) ⎪ 0 0 < ω < ωc ⎪ ϕ SSB (ω ) = ⎨ 0 −ωc < ω < 0 ⎪ ⎪⎩ Ac M (ω + ωc ) ω < −ω Demodulation of SSB Signals
SSB signals can be demodulated by one of two methods – coherent detection and envelope detection. Coherent Detection:
Regardless of the method by which the SSB signal is generated, the message signal is recovered by coherent detection. Consider that the local oscillator has a phase error of θ e . The output of the multiplier of the detector is x(t ) = Ac m(t ) cos(ωc t ) cos(ωc t + θ e ) + Ac mh (t ) sin(ωc t ) cos(ωc t + θ e ) A m(t ) A m(t ) A mˆ (t ) A mˆ (t ) cos(θ e ) + c cos(2ωc t + θ e ) + c sin(θ e ) − c sin 2ωc t + θ e ) = c 2 2 2 2
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After low-pass filtering we obtain y (t ) =
Ac m(t ) A mˆ (t ) cos(θ e ) + c sin(θ e ) 2 2
If the phase error is close to zero we obtain y (t ) = km(t ) , which is the desired message signal. If the phase error is close to π / 2 we obtain the Hilbert transform of the message signal. Therefore, the local oscillator must produce a negligible phase error. The Hilbert transform produces the Donald Duck sound effect. Envelope Detection of SSB Signals
In order to use envelope detection, we first insert a carrier with a very large amplitude at the receiver, that is,
ϕ SSB (t ) + Carrier = A0 cos(ωc t ) + m(t ) cos(ωct ) + mh (t ) sin(ωct ); A0 >> m(t )
ϕ SSB (t ) + Carrier = [ A0 + m(t )] cos(ωc t ) + mh (t ) sin(ωc t ) =
⎛ mh (t ) ⎞ + mh2 (t ) cos(ωc t + θ (t )); θ (t ) = − tan −1 ⎜ ⎟ ⎝ A0 + m(t ) ⎠
[ A0 + m(t )]
2
Since A0 >> m(t ) then
[ A0 + m(t )]
2
+ mh2 (t ) ≈ [ A0 + m(t )]
2
Therefore, we obtain ϕ SSB (t ) + Carrier ≈ [ A0 + m(t ) ] cos(ωc t + θ (t )) The above is essentially an AM signal, which can now be demodulated using an envelope detector. Vestigial Sideband (VSB) Modulation
Many message signals , such as TV video, Fax and high speed data have very large bandwidth a significant low-frequency content. Although SSB may be used to save bandwidth, they would require ideal sideband filters, which not realizable in practice. While DSB may be more suitable they require twice the message bandwidth for transmission. The best compromise between bandwidth savings and the required filtering is VSB. VSB is obtained by filtering DSB-SC or AM using a practical VSB filter. Rather than suppress an entire sideband, a trace or vestige (of width α Hz ) is passed by the filter as shown in Figure 16. The response of the VSB filter must have an odd symmetry about the carrier frequency f c
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or ωc . The VSB sideband has a transition interval of width 2α Hz . If the bandwidth of the baseband signal is B Hz then the bandwidth of the VSB signal is given by
BW = B + α ; α < B It is difficult to derive the time domain expression for the VSB. However, consider a sideband filter ( H SSB ( f ) ) to be an ideal high-pass filter and a vestige filter ( H α ( f ) ) as shown in Figure 16. Then we can consider the VSB filter to comprise the difference of the SSB filter and the vestige filter. That is, HVSB ( f ) = H SSB ( f ) − H α ( f ) If the output of the modulator is Ac [1 + m(t )]cos(ωc t ) (assuming µ = 1 ) then the output of VSB filter can be expressed as
ϕVSb (t ) = Ac cos(ωc t )+ Ac [ m(t ) cos(ωc t ) − mˆ (t ) sin(ωc t )]− Ac mα (t ) sin(ωc t )
VSB+carrier
carrier
vestige signal
SSB signal
VSB signal
If the carrier amplitude is large enough to avoid envelope distortion then an envelope detector can be used to recover m(t ) in the same way as was done for SSB modulation. Otherwise, coherent detection can be used to recover m(t ) .
Figure 16
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Frequency Conversion
Frequency conversion, also referred to as mixing is a process of translating a signal spectrum from one centre frequency to another. For example, modulation translates the message spectrum upwards in frequency to passband while demodulation translates downwards in frequency to baseband. Frequency conversion can also be used to translate a band-pass signal centred at one frequency to a new centre frequency. There are two types of frequency converters – up converters and down converters. Down converters translate a band-pass signal from a higher frequency to a lower frequency. Up converters, on the other hand, translate a signal from a lower frequency to a higher frequency. A block diagram of a frequency converter is shown in Figure 17. It consists of a local oscillator, mixer (the multiplier) followed by a filter. The filter determines whether the operation is down conversion or up conversion.
Figure 17
A block diagram for up or down conversion
Suppose vin (t ) = A1m(t ) cos(ω1t ) and vLo (t ) = A2 m(t ) cos(ω2t ) then the output of the mixer has two components, that is v1 (t ) = A1 A2 m(t ) cos(ω1t ) cos(ω2t ) =
1 1 A1 A2 m(t ) cos ( (ω2 − ω1 )t ) + A1 A2 m(t ) cos ( (ω2 + ω1 )t ) 2 2
For up conversion the filter is a band-pass filter centred at ω1 + ω2 while for down conversion it is centred at ω2 − ω1 . Superheterodyne Receiver
A simplified block diagram showing the basic elements is shown in Figure 18. The input from the antenna It consists of a combination of a tunable band-pass radio frequency (RF) filter and amplifier as the front end, a local oscillator, a mixer an intermediate frequency (IF) filter and amplifier for down conversion and a demodulator. The input from the antenna is a radio frequency signal at the carrier frequency of the form
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r (t ) = A(t ) cos(ωc t ) where A(t ) conveys the message signal. The local oscillator output is of the form vLO (t ) = 2 cos [ (ωc + ωIF )t ] ; ωIF = 2π f IF
The intermediate frequency is standard for all AM broadcast radios and is set to f IF = 455 kHz . The IF amplifier is a band-pass filter with a centre frequency of f IF = 455 kHz . The bulk of the amplification and filtering is done at the IF amplifier stage. It is much easier to design amplifiers and filters with high selectivity (ability to separate closely spaced signals) and high sensitivity (ability to detect very weak signals) at the fixed IF frequency. The dashed line connecting the two boxes denotes that the centre frequency of the RF filter and the local oscillator frequency are ganged and tuned together. That is, as f LO = f c + f IF is tuned (changed), the centre frequency of the RF filter f c is also changed so as to maintain the difference at f IF
Figure 17
A basic block diagram of the Superheterodyne receiver
The standard AM broadcast band extends from 540 kHz to 1600 kHz and so the RF filter is tunable from 540 kHz to 1600 kHz . Also, the local oscillator is tunable from f LO =995 kHz to 2055 kHz 540. Image Frequency
An image frequency is an input radio frequency at fim = f c + 2 f IF . It is called an image frequency because, when mixed the local oscillator signal, it will produce a signal at f IF in the same way as the desired signal; see Figure 18. Normally, the RF filter would suppress it before it enters the mixer.
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Figure 18
Illustration of image frequency
Carrier Acquisition
For coherent detection the frequency and phase of the modulated signal must be very close to the phase angle of the local oscillator. To accomplish coherent detection one can either track and acquire the phase or regenerate the phase at the receiver. There are two common methods for tracking or generating the phase - the phase-locked loop (PLL) method and the Costas loop method. The Phase-Locked Loop (PLL) Method
A PLL is a feedback network (as shown in Figure 19) that is capable of tracking and acquiring the frequency and phase of a modulated signal. It consists of three basic elements: 1. A voltage-controlled oscillator (VCO) 2. A phase comparator comprising a multiplier 3. A loop (low-pass) filter A regular oscillator runs freely without the need for an input; its output is ωc . A VCO is an oscillator whose output frequency and phase are controlled by an input voltage. Suppose the input voltage to a VCO is e0 (t ) . The output frequency is a linear function of the input voltage, that is,
ω (t ) = ωc + ce0 (t ) The constant of proportionality is c and ωc is the free-running frequency when the input is zero. The two inputs to the multiplier must always be in phase quadrature. That is, if modulated
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carrier input is a sine then the output of the VCO must be cosine and vice versa. This is done so that whenever the two phases are equal, the output of the low-pass (loop) filter is zero and the phase of the VCO stops changing denoting that the loop is in lock.
Figure 19 Phase-Locked Loop With the inputs as shown in Figure 19, the output of the multiplier is x(t ) = AB sin(ωc t + θ i ) cos(ωct + θ o ) =
AB AB sin(θ i − θ o ) + sin(2ωc t + θ i + θ o ); θi ≠ θ o 2 2
The low-pass filter output is eo (t ) =
AB sin(θ e ); θ e = θi − θ o 2
We see that when θi = θ o , that is, the phase error θ e = 0 , then the input to the VCO is also zero and the output phase stays unchanged. If θ e ≠ 0 then the corresponding nonzero voltage eo (t ) will control the VCO to change its phase in a direction that forces the phase error to be zero. This way, the PLL tracks the phase of the sinusoidal input. Definitions:
(a) The PLL is said to be in phase-lock mode when the two sinusoidal inputs to the multiplier have the same phase or in phase coherence ( θ e = 0 ) (b) When in the lock mode, the PLL can only track changes within a finite range. The holdin or lock range is the range of incoming phases (or frequencies) that the PLL can still track. Outside the hold-in range the loop diverges and never return to the lock mode
(c) If the PLL is not in a lock mode only a finite range of incoming phases (or frequencies) can cause the loop to go into a lock mode. The pull-in or capture range is the range of changes that will cause the loop to go into a lock mode The PLL is not capable of tracking phase or frequency changes that are too rapid. The PLL can also be used to demodulate angle modulated signals such as frequency and modulated signals.
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Costas Loop
The Costas loop accomplishes phase tracking and demodulation of DSB-SC and SSB signals simultaneously. The basic block diagram of the Costas loop is shown in Figure 20.
Figure 20 Costas Loop Initially, the VCO of the Costas loop generates a local carrier 2 cos(ωct + θ o ) , whose phase differs from that of the incoming modulated carrier m(t ) cos(ωc t + θi ) . A second local carrier 2sin(ωc t + θ o ) is formed by adding a −π / 2 . The two (cosine and sine) local carriers are used to coherent detect the incoming signal. The two modulated signals (top and bottom rails) are given by xc (t ) = m(t ) cos(θ e ); xs (t ) = m(t ) sin(θ e ); θ e = θi − θ o The two demodulated signals xs (t ) and xc (t ) are multiplied to yield the input to the low-pass filter y (t ) = m 2 (t ) sin(θ e ) cos(θ e ) =
1 2 m (t ) sin(2θ e ) 2
The bandwidth of the low-pass is so narrow that it only extracts the dc component. Therefore, the output of the low-pass filter is K sin(θ e ) . This output is used to control the phase of the VCO to force the phase error to zero. When the phase error is zero the VCO phase stays unchanged. Then, xs (t ) = 0 and xc (t ) = m(t ) is the demodulated signal. When a change occurs in the incoming phase then the phase error θ e , also changes. The change causes the VCO phase to also change in a direction that forces the phase error to zero. This way, the Costas loop tracks and demodulates the incoming DSB-SC, SSB or AM signal.
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