Communication

Systems

Chapter 4
Continuous Wave Modulation
Linear CW Modulation
Dr. Le Dang Quang
Department of Telecommunications (113B3)
Ho Chi Minh City University of Technology
Email: ldquang@hcmut.edu.vn

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 Chapter Outline

4.1 Modulation types

4.2 AM
4.3 DSB
4.4 SSB, VSB, VSB+C
4.5 Frequency Conversion & Linear Demodulation

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4.1 Modulation types

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 Modulation Types
Modulation is the systematic alteration of one waveform, called the
carrier, according to the characteristics of another waveform, the
modulating signal or message. The goal is to produce an information
bearing modulated waveform best suited to the given communication task.

Analog modulation methods can be divided into 2 main classes:

continuous-wave modulation and pulse coded modulation. Furthermore,
1. CW-modulation (continuous-wave) can be:
a) Linear CW-modulation
AM amplitude modulation
DSB double sideband modulation
VSB vestigial sideband modulation
SSB single sideband modulation

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 Modulation Types
b) Exponential CW-modulation
FM frequency modulation
PM phase modulation

2) Pulse coded modulation (for discrete signals) can be:

PAM pulse coded amplitude modulation
PDM (PWM) pulse delay modulation
PPM pulse position modulation

CW modulation includes a high frequency sinusoidal signal, pulse

modulation contains a square wave.

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 Modulation Types
❑ Example of Analog Modulation Methods

E.g., applications of AM signals: commercial broadcasting,

multiplexing of telephone signals, short wave military and amateur
communications.
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4.2 Amplitude Modulation

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 Linear CW Modulation
❑ Assumptions:
Assumptions of information signal x(t) are as follows
▪ Bandlimited:

▪ Normalized:

Tone Modulation:

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 AM Modulation
❑ AM-Signal in time domain:

where fc is carrier frequency,

Ac is the amplitude of the
unmodulated carrier,
μ > 0: modulation index.

AM-signal (modulated
envelope):

Envelope reproduces the shape

of x(t) if

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 AM Modulation
▪ 100 % modulation: μ = 1

▪ Overmodulation: μ > 1: It
causes phase reversals and
envelope distortion.

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 AM Modulation
❑ AM-Signal in Frequency Domain

Here, we have written out only the positive-frequency half. Negative

frequency half will be the hermitian image of the above equation.

The bandwidth of the modulated signal is twice the bandwidth of

modulating (message) signal: BT = 2W.

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 AM Modulation
❑ Power of AM-Signal:
Average transmitted power is:

When fc >> W, second term averages to zero. If

and

Then,

This can be represented by using the power of unmodulated carrier Pc

and the power per sideband Psb:

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 AM Modulation
It follows from the condition |μ x(t)| ≤ 1 that

Consequently, at least 50% of the total transmitted power resides in

a carrier term that’s independent of x(t) and thus conveys no message
information.

AM-signal is not practical for transmitting DC-component or signals

with significant low frequency content.

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4.3 Double-SideBand
suppressed-carrier Modulation

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 DSB-Modulation
❑ DSB: double-sideband suppressed-carrier modulation: two
sidebands, suppressed carrier

▪ In frequency domain:

Like AM-spectrum without carrier

Bandwidth BT = 2W
Phase reversal whenever x(t) crosses 0.

▪ In Time domain:
Envelope:

Phase:

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 DSB-Modulation
Note: Envelope itself cannot be used for detection because of the
phase reversal
• Detection is more difficult than in the case of AM
• DSB conserves power but requires complicated
demodulation circuitry, whereas AM requires increased
power to permit simple envelope detection.

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 DSB-Modulation
❑ Power of DSB-signal:
All of the average transmitted power is used for information transition.

The peak envelope power is the limiting factor in the transmitters

Amax2 (Amax = max A(t) ).

If Amax2 is fixed and other factors are equal, a DSB-transmitter

produces four times the sideband power of an AM-transmitter. Since
the transmission range is proportional to the power per sideband, DSB
provide 4 times longer path length.

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 Tone Modulation and Phasor Analysis
Setting

DSB:

AM:

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 Tone Modulation and Phasor Analysis
Phasor analysis is helpful for studying effects of transmission
distortion and interference:

Example: Consider tone-modulated AM with Am = 2/3. The phasor

diagram is constructed as (including carrier phasor and two sideband
phasors):

The phasor sum equals the envelope Ac(1 + 2/3 cosmt)

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 Tone Modulation and Phasor Analysis
Suppose that a transmission channel completely removes the lower
sideband, then we obtain a phasor as:

Now the envelope becomes:

Therefore, the envelop distortion can be determined. Also, the

amplitude distortion produces a time-varying phase (t).

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 Ví dụ

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21 Thanh Tuấn
Th.S. Nguyễn
 Ví dụ

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 Ví dụ

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4.4 SSB, VSB, VSB+C

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 SSB-Modulation
❑ SSB: Single-sideband (suppressed-sideband)

AM is wasteful of both power and bandwidth, DSB is wasteful of

bandwidth.

In the case of real baseband signals, the positive frequencies contains

all the information of the signal.

=> The upper and lower sidebands of DSB are symmetric about the
carrier frequency, so either one contains all the message information.

USSB: Lower sideband is removed.

LSSB: Upper sideband is removed.

In both cases, the bandwidth is

and the transmitted power is

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 SSB-Modulation
SSB spectra:

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 SSB-Modulation
Tone-modulation:

Note: Pure sine wave with frequency fc ± fm

Envelope is constant => envelope detection does not work.

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 Hilbert Transform
Hilbert transform is useful when analysing the SSB modulation. We
will also see in what follows the relationship between Hilbert
transform and analytic signals.

In the following, we are using a quadrature filter with transfer

function:

The corresponding impulse response is

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 Hilbert Transform
Now, let the input signal to a quadrature filter be x(t). Then, the output
signal 𝑥(𝑡)
ො is defined to be the Hilbert-transform of x(t):

The quadrature filter is non-causal, so it is not realizable. It can be,

however, well approximated over a finite frequency band.

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 Hilbert Transform
❑ Hilbert-transform – properties:

▪ A signal x(t) and its Hilbert transform 𝑥(𝑡)

ො have the same
amplitude spectrum. In addition, the energy or power in a signal
and in its Hilbert transform are also equal.

▪ If 𝑥(𝑡)
ො is the Hilbert transform of x(t), then - x(t) is the Hilbert
transform of 𝑥(𝑡).
ො i.e., that two successive frequency shifts of -90
degrees result in a total shift of –180 degrees.

▪ A signal x(t) and its Hilbert transform 𝑥(𝑡)

ො are orthogonal.

▪ Hilbert transforms are useful in analyzing SSB signals.

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 Hilbert Transform
Examples:
(a) The Hilbert-transform for a sinusoidal signal corresponds to 90°
phase-shift:

(b) The Hilbert-transform of a rectangular pulse is:

Then,

(See more on p. 123, [1])

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SSB Modulation: Analysis with Arbitrary Message
We begin with the general DSB-modulated signal:

The idea is to use the method of low-pass equivalent signals:

We assume that the sideband filters are ideal:

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SSB Modulation: Analysis with Arbitrary Message

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SSB Modulation: Analysis with Arbitrary Message
Then, the output of the ideal filter:

Frequency domain waveform of SSB signal:

Time-domain waveform of SSB signal:

A signal under the form is called an analytic signal.

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SSB Modulation: Analysis with Arbitrary Message
Lowpass – bandpass transformation yields:

This means that we can represent the SSB-modulated signal with in-
phase and quadrature-phase components as:

The SSB envelope is:

Due to the complexity of the above expression, it is a difficult task to

sketch SSB waveforms or to determine the peak envelope power.

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SSB Modulation: Analysis with Arbitrary Message
Remarks:

▪ Whenever the SSB modulating signal has abrupt transitions, the

Hilbert transform contains sharp peaks. In practice, the message
needs to be lowpass filtered before SSB modulation.

▪ SSB is not appropriate for pulse transmission, digital data, or

similar applications, and more suitable modulating signals (such as
audio waveforms) should still be lowpass filtered before
modulation, in order to smooth out any abrupt transitions that
might cause excessive horns or smearing.

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 Analytic Signal
An analytic signal, in general, is a signal which has only positive
frequency components. Thus, its frequency spectrum is zero when f < 0.

An analytic signal is also defined as:

(this is also called pre-envelope of a real signal).

Analytic signals are used in the generation of SSB modulation. SSB

modulation can be studied as follows: first, we form the analytic lowpass
signal, then we apply a frequency translation (at carrier frequency):

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 VSB Modulation
VSB (Vestigial sideband): A compromise between DSB and SSB:
▪ Bandwidth closer to SSB
▪ Easier to implement than SSB
▪ The message can include also small frequencies

Consider a modulation signal of very large bandwidth having significant

low frequency content (e.g., television video, facsimile, high-speed data
signals). Bandwidth conservation argues for the use of SSB, but
practical SSB signals have poor low-frequency response (due to non-
ideality of practical filters). On the other hand, DSB works quite well for
low message frequencies, but the transmission bandwidth is twice that of
SSB. The compromise is VSB.

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 VSB Modulation
One sideband is passed almost completely, while just a trace, or a
vestige, of the other sideband is included.

If we take the USB case, the sideband filter transfer function H(f) has
to fulfil:

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 VSB Modulation
The filter transition band is symmetric with respect to fc in a way that:

Usually, we want that the transfer function HT(f) of the total chain
(transmitter + receiver) is symmetric in a way that:

The processing of the vestigial sideband can be done either in the

transmitter or in the receiver. Then, this calls for a little bit wider
vestigial sideband in the other end. The processing can also be shared
equally between the transmitter and the receiver.

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 VSB Modulation
❑ VSB signal in time domain

The transmitted power is not easy to be determined exactly but it is

bounded by:

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 VSB Modulation
❑ VSB+C (VSB & carrier)
Time domain waveform:

In-phase and quadrature-phase components are given by

The envelope is

Hence, if μ is not too large and β not too small, then

 Envelope detection works.

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 VSB Modulation

In practice, we have to find 𝜇, 𝛽 to compromise between the following:

▪ Envelope distortion (related to the demodulation complexity)
▪ Power efficiency
▪ Bandwidth

Usually, practical solutions are found empirically.

In practice, VSB+C is used in traditional TV broadcasting systems,

such as NTSC, PAL and SECAM.

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 I/Q Modulation or QAM
QAM: Quadrature Amplitude Modulation.

The general representation of a bandpass signal consists of two independent

components (in-phase and quadrature-phase components). Based on this,
it is possible to modulate two independent messages, x1(t) and x2(t), into
one carrier => QAM signal:

Assuming that the bandwidths of the two messages are the same, the
bandwidth of the QAM signal equals the DSB bandwidth. Then, if it is
necessary to transmit two messages, QAM has the SSB bandwidth
efficiency.
This QAM modulation principle is used commonly as follows:
▪ PAL and NTSC colour TV systems are partially based on QAM
▪ QAM is used extensively in digital communications (will be studied
later)
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4.5 Frequency Conversion &
Linear Demodulation

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Frequency Conversion and Linear Demodulation

Demodulation (for AM, DSB, SSB and VSB) implies downward

frequency translation in order to recover the original message.

Demodulators fall into two categories:

▪ Envelope (non-coherent) detectors.
▪ Synchronous (coherent) detectors.

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 Envelope Detection (for AM)

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 Synchronous Detection
The local oscillator signal is synchronized in phase and frequency with
the carrier.

A generalized AM-/DSB-/SSB-/VSB-signal:

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 Synchronous Detection
Here, KD is a detector specific constant. If needed, the DC-term can be
filtered out.

It should be noted that the synchronous detector attenuates the

quadrature-phase component completely. Hence, effectively
remove ½ noise power (discuss latter).

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 Synchronous Detection of a VSB Signal

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 Carrier Synchronization
In a good-quality synchronous detection, the local oscillator signal is
exactly synchronized to the carrier in both frequency and phase. The
synchronization can be based on:

1. Carrier (if present, the envelope detection is usually used).

2. Partially attenuated carrier.
3. Pilot-signal that is synchronized to the carrier (e.g., half of th
carrier frequency).
4. Short carrier burst that is repeated from time to time.

In practice, a phase-locked loop (PLL) is used to lock the local

oscillator phase and frequency to the received pilot-signal.

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Phase & Frequency Errors in Synchronous Detection
The local oscillator signal:

DSB with sine wave modulation – demodulated signal:

SSB with sine wave modulation – demodulated signal:

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Phase & Frequency Errors in Synchronous Detection
If the LO phase is drifting: φ‘ ≠ 0, ω‘ = 0
▪ with SSB, the phase errors appear as delay distortion (extreme
case: delay of 90 degrees  output signal becomes the Hilbert
transform of the input signal). However, human ear can tolerate
rather high delay distortions  no serious effect in speech signals
▪ with DSB, the amplitude is varying 0 ... KD (if phase error is + or –
90 degrees, the amplitude vanishes completely)  an apparent
fading effect.
If the LO frequency is drifting: ω‘ ≠ 0, φ‘ = 0
▪ with SSB the frequency is changing  the harmonic structure of
speech is distorted, tolerable error ± 10 Hz.
▪ with DSB, a pair of frequency tones is produced  worse than with
SSB.

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Phase & Frequency Errors in Synchronous Detection
Summary: phase and frequency synchronization requirements are rather
modest for voice transmission via SSB. But in data, facsimile and video
systems with suppressed carrier, careful synchronization is a must.
Consequently, TV broadcasting employs VSB+C rather than suppressed
carrier VSB.

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 Synchronization Errors in QAM
In the case of QAM, the in-phase and quadrature modulated signals can be
separated only if the local oscillator is well synchronized to the carrier.

Phase error in local oscillator causes leakage. For instance, 900 phase
error changes the I to Q signal and vice versa.

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 Bài tập 1

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HCMUT
 Bài tập 2

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 Bài tập 3

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 Bài tập 6

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 Bài tập 7

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 Bài tập 8

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 Bài tập 9

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 Bài tập 10

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 Bài tập 11

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 Bài tập 13

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 Bài tập 14

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 Bài tập 15
❑Cho tín hiệu sau điều chế có phổ biên độ (tần số
dương) |Y(F)| = (F – 8) + 5(F – 10) + (F – 12)
(F:KHz). Xác định 1 biểu thức thích hợp của tín
hiệu cần điều chế (thỏa điều kiện chuẩn hóa) và
sóng mang trong các trường hợp sau:
a) Điều chế biên độ AM.
b) Điều chế hai biên triệt sóng mang DSB.
c) Điều chế biên trên USSB.
d) Điều chế biên dưới LSSB.

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 Bài tập 16
❑Cho phổ biên độ (tần số dương) của tín hiệu
sau điều chế có dạng |Xc(F)| = 10(F – 48) +
99(F – 50) + 10(F – 52) (F:Hz). Xác định
1 loại điều chế tương tự kèm thông số thích
hợp cùng biểu thức của sóng mang (có biên
độ và tần số càng nhỏ càng tốt) và tín hiệu
thông tin chuẩn hóa cần điều chế.

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 Bài tập 17
❑ Cho tín hiệu đơn tần cần điều chế x(t) = 0.8cos4πt (t:ms)
và sóng mang 10sin20πt (t:ms).
a) Tín hiệu x(t) được điều chế biên độ (AM) với chỉ số điều
chế μ = 0.5. Vẽ dạng sóng của tín hiệu sau điều chế.
b) Tín hiệu x(t) được điều chế hai biên triệt sóng mang
(DSB). Tính công suất của tín hiệu sau điều chế.
c) Tín hiệu x(t) được điều chế biên trên (USSB). Viết biểu
thức (theo thời gian) của tín hiệu sau điều chế.
d) Thiết kế 1 sơ đồ nguyên lý thực hiện điều chế DSB từ các
bộ điều chế AM (chỉ số điều chế µ, bộ tạo sóng mang
nằm trong bộ điều chế), bộ khuếch đại và bộ cộng.

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 Bài tập 18
❑ Cho sóng mang có biểu thức 10cos10πt (t:ms).
a) Cho tín hiệu x(t) = cos2πt (t:ms) được điều chế biên độ (AM)
80% rồi đưa qua kênh truyền có suy hao toàn tuyến là 20dB.
Tính tỉ số công suất tín hiệu trên nhiễu tại ngõ vào bộ lọc thu lý
tưởng, giả sử nhiễu trên kênh truyền được mô hình như AWGN
với hàm mật độ phổ công suất hai phía Gn(f) = N0/2 =10-
10W/Hz.

b) Cho tín hiệu x(t) = sin22πt (t:ms) được điều chế hai biên triệt
sóng mang (DSB). Vẽ dạng sóng và vẽ phổ biên độ (tần số
dương) của tín hiệu sau điều chế.
c) Cho tín hiệu x(t) = 0.5cos2πt + 0.5sin2πt (t:ms) được điều chế
biên dưới (LSSB). Xác định biểu thức (theo thời gian) và tính
công suất của tín hiệu sau điều chế.

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 Bài tập 19
❑ Cho sóng mang có biểu thức 10cos10πt (t:ms).
a) Cho tín hiệu x(t) = cos2πt (t:ms) được điều chế biên độ (AM)
80% rồi đưa qua kênh truyền có suy hao toàn tuyến là 20dB.
Tính tỉ số công suất tín hiệu trên nhiễu tại ngõ vào bộ lọc thu lý
tưởng, giả sử nhiễu trên kênh truyền được mô hình như AWGN
với hàm mật độ phổ công suất hai phía Gn(f) = N0/2 =10-
10W/Hz.

b) Cho tín hiệu x(t) = sin22πt (t:ms) được điều chế hai biên triệt
sóng mang (DSB). Vẽ dạng sóng và vẽ phổ biên độ (phía tần số
dương) của tín hiệu sau điều chế.
c) Cho tín hiệu x(t) = 0.5cos2πt + 0.5sin2πt (t:ms) được điều chế
biên trên (USSB). Xác định biểu thức đầy đủ (theo thời gian)
và tính công suất của tín hiệu sau điều chế.

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 Bài tập 20
❑ Cho hệ thống thu vô tuyến quảng bá đổi tần AM có các
thông số sau: tần số sóng mang fc của mỗi đài AM thay
đổi từ 540KHz đến 1600KHz với khoảng cách giữa hai
sóng mang liên tiếp là 10KHz.
a) Trong trường hợp tần số trung tần fIF = 455KHz và tần số
bộ dao động nội fLO > fc, xác định phạm vi thay đổi của
tần số bộ dao động nội FLO để có thể thu tối đa các đài
AM và xác định phạm vi ảnh hưởng của tần số ảnh.
b) Tìm điều kiện của tần số trung tần fIF để phạm vi tần số
ảnh nằm ngoài toàn bộ băng tần AM.

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Faculty of EEE 75 Nguyễn Thanh Tuấn
Th.S.
HCMUT
Câu 4 (3đ): Cho tín hiệu đơn tần cần điều chế
x(t) = 0.4cos4πt (t:ms) và sóng mang 10sin40πt (t:ms).

a. Tín hiệu x(t) được điều chế biên độ (AM) với hệ số điều chế μ=2. Vẽ
dạng sóng của tín hiệu sau điều chế.
Tìm điều kiện của hệ số điều chế để có thể sử dụng bộ tách sóng đường
bao trong trường hợp cụ thể với tín hiệu và sóng mang như trên.

b. Tín hiệu x(t) được điều chế hai biên triệt sóng mang (DSB). Tính công
suất của tín hiệu sau điều chế.
c. Tín hiệu x(t) được điều chế biên trên (USSB). Vẽ phổ của tín hiệu sau
điều chế.

d. Tín hiệu x(t) được điều chế tần số (FM) với độ di tần fΔ= 5KHz. Vẽ phổ
biên độ (tần số dương) của tín hiệu sau điều chế.

e. Tín hiệu x(t) được điều chế pha (PM) với độ di pha φΔ= 100o. Ước
lượng băng thông truyền BT theo tiêu chuẩn Carson của tín hiệu sau điều
chế.
Telecomm. Dept. CS-2016
Faculty of EEE 77 HCMUT