BEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS

(BUAA)

TASK : FIRST ASSIGNMENT

CHINESE NAME : 布瑞西

ENGLISH NAME : ABU YUSIF KARGBO

STUDENT NUMBER : LS2402204

SCHOOL : ELECTRONIC, INFORMATION ENGINEERING

COURSE : INFORMATION AND COMMUNICATION ENGINEERING

CATEGORY : MASTERS PROGRAM

Question 2
Solutions
Given the following:
1. Autocorrelation: Rx(τ) = 𝑒 −10000|𝜏|
20000
2. 𝑃𝑆𝐷: SX(f) = (using the Fourier transform of exponential
(10000)2 + (2𝜋𝑓)2

autocorrelation)
3. filter (ideal low pass with cut-off frequency) = fco
4. signal sample = fs
5. 16 level quantizer = L = 16 ⇒ log2(16) = 4bits/sample
6. quantizer range[ − 𝑉𝑚𝑎𝑥 , 𝑉𝑚𝑎𝑥 ], (𝑉𝑚𝑎𝑥 = 4𝜎𝑋 )
7. 8 PSK Modulation = 3bits/symbol

A. Determine the cut-off frequency of fco, such that the filtered signal preserved 90%
of its total power before filtering.
20000
SX(f) = (10000)2 + (2𝜋𝑓)2

∞ 𝑓 20000
Total power = ∫−∞ 𝑆𝑋 (𝑓) 𝑑𝑓 = ∫−𝑓𝑐𝑜 =1
𝑐𝑜 2.10000

1 𝜋
(using standard results = 2 𝑑𝑥 = 𝑎 )
𝑎2 + 𝑥 ′

𝑓
To preserve 90% of power, solve for fco = ∫−𝑓𝑐𝑜 𝑆𝑋 (𝑓) 𝑑𝑓 = 0.9
𝑐𝑜

𝑓 20000
∫−𝑓𝑐𝑜 2 2 df = 0.9 change the variables to 𝑢 = 2𝜋𝑓 ⇒ 2𝜋𝑑𝑓
𝑐𝑜 (10000) + (2𝜋𝑓)

20000 2𝜋𝑓𝑐𝑜 20000 20000 1 −1

𝑢 2𝜋𝑓𝑐𝑜
∫ 2 2
𝑑𝑢 = ⋅ ⋅ 𝑡𝑎𝑛 ( ) ⎹−2𝜋𝑓
2𝜋 −2𝜋𝑓𝑐𝑜 (10000) + (2𝜋𝑓) 2𝜋 10000 10000 𝑐𝑜

20000 2𝜋𝑓 2𝜋𝑓 2 2𝜋𝑓

⋅ [𝑡𝑎𝑛−1 (10000
𝑐𝑜
) − ⋅ 𝑡𝑎𝑛−1 (10000
𝑐𝑜
)] = ⋅ 𝑡𝑎𝑛 −1 (10000
𝑐𝑜
)
2𝜋 ⋅10000 𝜋

Set equal to 0.9

2 2𝜋𝑓𝑐𝑜 2𝜋𝑓𝑐𝑜 2
⋅ 𝑡𝑎𝑛 −1 ( ) = 0.9 ⇒ 𝑡𝑎𝑛 −1 ( ) = 0.9 ⋅ = 1.4137𝑟𝑎𝑑
𝜋 10000 10000 𝜋

2𝜋𝑓𝑐𝑜 6.1787 ⋅1000

= tan(1.4137) ≈ 6.1787 ⇒ 𝑓𝑐𝑜 = 𝑓𝑐𝑜 ≈ 9840𝐻𝑧
10000 2𝜋

B. Determine the minimum value of the sampling frequency such that the sampled
signal does not experience any aliasing.
𝑓𝑐𝑜 ≥ 2 ⋅ 𝑓𝑐𝑜 = 2.9840𝐻𝑧
Therefore, 𝑓𝑠 = 19680Hz (minimum)
 C. Calculate the average signal power to the average quantization error variance (S/N)q
in [dB]. Neglect the effect of quantization noise due to saturations.
Firstly, do the quantization of noise power
2
2
𝑉𝑚𝑎𝑥 (4𝜎𝑋 )2 16𝜎𝑋 𝜎2
𝜎𝑞2 3𝐿2
𝑤ℎ𝑒𝑟𝑒 𝑉𝑚𝑎𝑥 = 4𝜎𝑋, 𝐿 = 16 ⇒ 𝜎𝑞2 = 3.162
= 768
= 48𝑋

2
𝑆 𝜎𝑋
= 2 = 48 𝑆𝑁𝑅(𝑑𝐵) = 10𝑙𝑜𝑔10 (48) ≈ 16.8𝑑𝐵
𝑁𝑞 𝜎𝑋 /48

𝑆
(𝑁 )𝑑𝐵 ≈ 16.8𝑑𝐵
𝑞

D. Calculate the values of bit rate and symbol rate.

Firstly: Bits/sample = 4
𝑅𝑏 = 𝑓𝑠 . 𝑏𝑖𝑡𝑠/𝑠𝑎𝑝𝑙𝑒𝑠 = 19680 ⋅ 4 = 78720bits/sec.

Secondly: symbol rate (8-PSK)

Sampling rate 𝑓𝑠 = 19680𝑠𝑎𝑚𝑝𝑙𝑒𝑠/𝑠𝑒𝑐.
𝑅 78720
𝑏
symbol rate = 𝑏𝑖𝑡𝑠/𝑠𝑦𝑚𝑏𝑜𝑙 = = 26240symbols/sec.
3

Question 1
A. Draw the block diagram of communication system.

Voice
Email Output
Transmitted
Signal Transmission Signal
Signal

Input Received
Signal Signal
Message

Message
Output
Input

Input Transmitter Channel Receive Output

Transduce r Transduce

Modulation
Demodulation
Analog/Digital
Decoding
conversion
Digital/Analogue conversion
Microphone
Camera Noise and distortion
keyboard
B. What parameters to describe random signals common in communications?
1. THE MEAN
The mean of a random signal is a basic statistical indicator that represents the
average value of the signal over time. It is shown as E[X] and pronounced as
∞
follows: 𝐸 [𝑋] = ∫−∞ 𝑥𝑓𝑋 (𝑥)𝑑𝑥 .where 𝑓𝑋 (𝑥) is the probability density function

(PDF) of the signal. The mean indicates the central tendency of a signal, showing
where its values cluster. In communication systems, it helps identify the signal’s
baseline level, aiding in the detection of noise or interference. A non-zero mean can
affect receiver thresholds and influence modulation schemes, such as determining
the carrier level in AM. Practically, the mean is estimated using sample averages
1
from observed data. 𝐸 [𝑋] = ∑𝑁
𝑖−1 𝑋𝑖 where N is the number of samples. The
𝑁

mean alone does not convey the variability or spread of a signal. Therefore, it is
typically considered alongside parameters like variance. But the mean is a
foundational metric for analyzing random signals and guiding signal processing
decisions in communication systems.
2. VARIANCE
This measures the dispersion (spread) of a random signal around its mean, denoted
as Var(X) or σ2. This quantifies how much the values of the signal deviate from the
mean on average. The formula for variance is given by: 𝐸 [𝑋] = 𝐸 [(𝑋 − 𝐸[𝑋])2 =
∞
∫−∞(𝑥 − 𝜇 )2 𝑓𝑋 (𝑥)2 𝑑𝑥.

where μ=E[X] is the mean of the signal. Variance measures how much signal values
deviate from the mean, with high variance indicating a wide spread and low
variance suggesting tight clustering. In communication systems, it is crucial for
evaluating the reliability of signal transmission. High variance can lead to increased
interpretation errors, affecting data accuracy. Engineers use variance to design
systems that handle expected signal fluctuations effectively. In adaptive modulation,
variance guides the selection of appropriate modulation techniques based on
channel conditions. Additionally, variance plays a key role in determining the
 signal-to-noise ratio (SNR), influencing the system's noise floor.
1
𝑉𝑎𝑟 (𝑋) = ∑𝑁 2
𝑖−1 𝑋𝑖 (𝑋𝑖 − 𝐸[𝑋]) where E[X] is the sample.
𝑁−1

3. POWER SPECTRAL DENSITY

This describes how a random signal’s power is distributed across its frequency
components. It is crucial in communications for understanding a signal’s frequency
characteristics. The PSD is typically represented as 𝑆𝑥 (𝑓) , where f denotes
frequency. The PSD can be obtained from the Fourier transform of the
autocorrelation function 𝑅𝑥 (𝜏) using the Wiener-Khinchin theorem: 𝑆𝑥 (𝑓) =
𝐹 { 𝑅𝑥 (𝜏)}.
Power Spectral Density analysis reveals how a signal’s power distributes across
frequency components and provides critical guidance for tasks like filtering,
modulation, and bandwidth allocation. The shape of the PSD indicates whether a
signal is wideband or narrowband, which influences antenna and
transmission-media design. PSD estimation is typically performed using techniques
such as the periodogram and Welch’s method. It also allows engineers to quantify
the impact of noise by showing how noise power overlaps with the signal. Signal
bandwidth is often defined by the range of frequencies where the PSD exceeds a
specified threshold. Consequently, the PSD remains a fundamental tool for
frequency-domain analysis and communication system design.
4. AUTOCORRELATION FUNCTIONS
This function measures how a random signal correlates with itself over different
time lags and is defined mathematically as:
∞
𝑅𝑥 (𝜏) = 𝐸[𝑋 (𝑡) 𝑋(𝑡 + 𝜏)] = ∫−∞ 𝑥(𝑡)𝑥(𝑡 + 𝜏 ) 𝑓𝑋 (𝑥)𝑑𝑥

The autocorrelation function reveals the temporal structure of a signal, exposing

hidden patterns and periodicities. It is vital in communication systems for analyzing
how signals evolve over time. High autocorrelation at specific lags indicates
similarity between signal values at different time points, aiding in prediction. A
rapid decay in autocorrelation suggests a wide signal bandwidth. This function
helps engineers design filters that retain desired signal components while
 suppressing noise.
Autocorrelation is also used to estimate channel characteristics in wireless systems,
supporting adaptive equalization. It plays a role in detecting signal stationarity, as
stationary signals exhibit consistent autocorrelation over time. Understanding
autocorrelation assists in optimizing system performance under time-varying
conditions. Overall, it is a key tool for analyzing the temporal behavior of random
signals in communication systems.
5. PROBABILITY DISTRIBUTION FUNCTION (PDF)
This describes the likelihood of a random signal taking on a specific value. It is
denoted as 𝑓𝑥 (𝑥) and provides a complete statistical characterization of the
∞
signal's distribution. The PDF must satisfy the property: ∫−∞ 𝑓𝑋 (𝑥)𝑑𝑥 = 1. The

probability density function (PDF) helps evaluate how noise and interference affect
signal performance in communication systems. It also quantifies the likelihood that
a signal falls within a specific value range by measuring the area under the curve:
𝑏
𝑃(𝑎 < 𝑋 < 𝑏) = ∫𝑎 𝑓𝑋 (𝑥)𝑑𝑥

For continuous signals, the probability density function (PDF) must integrate to one
over the full range of values. Understanding its shape is essential for selecting
suitable modulation schemes and error-correcting codes. In practice, the PDF can
be estimated from data using methods like kernel density estimation. It is also used
in simulations to model random signals and evaluate system performance under
realistic conditions. Overall, the PDF is a fundamental tool for characterizing
random signals and guiding communication system design.
6. CUMULATIVE DISTRIBUTION FUNCTION (CDF)
The CDF represents the probability that a random signal will take a value less than
or equal to a specific value xx. It is denoted as 𝑓𝑋 (x) and is derived from the PDF
𝑏
by integrating the PDF over the desired range: 𝐹𝑋 (𝑥) = ∫𝑎 𝑓𝑋 (𝑥)𝑑𝑥 . The

cumulative distribution function (CDF) offers a comprehensive view of a signal’s

distribution, helping engineers evaluate the likelihood of different signal levels. In
communication systems, it is essential for assessing performance under noise and
 interference and for computing probabilities of signal levels exceeding specific
thresholds: 𝑃(𝑋 > 𝑥) = 1 − 𝐹𝑋 (𝑥)
The cumulative distribution function (CDF) is essential for designing reliable
communication links and evaluating the performance of modulation schemes and
error-correcting codes. It provides insights into signal reliability and the impact of
noise, aiding in system optimization. Engineers can estimate the CDF from sample
data or use it in simulations to model random signal behavior under various
scenarios. Overall, the CDF is a key tool for characterizing the statistical behavior
of random signals in communication systems.
7. STATIONARITY
Stationarity means that a random signal’s statistical properties, like mean, variance,
and autocorrelation, remain constant over time. A process is weakly stationary if
these properties are time-invariant, simplifying analysis and prediction 𝐸 [𝑋(𝑡)] =
𝜇 and 𝑉𝑎𝑟 (𝑋 (𝑡) = 𝜎 2 ∀𝑡 . In communication systems, stationarity is a key
assumption that simplifies the analysis and design of signal processing techniques.
Non-stationary signals, with time-varying statistical properties, require more
advanced methods for analysis. Stationarity is classified into strict and weak forms,
with weak stationarity—requiring constant mean and variance—being more
common in practice. Understanding a signal's stationarity affects decisions on
modulation, filtering, and equalization. Engineers often use methods like
windowing or detrending to render non-stationary signals stationary for effective
analysis.
8. GAUSSIANITY
Gaussianity refers to whether a random signal follows a Gaussian distribution,
characterized by its bell-shaped curve. In communication systems, many signals
approximate this distribution due to the Central Limit Theorem
1 2 /2𝜎 2
𝐹𝑋 (𝑥) = 2
𝑒 −(𝑥− 𝜇 where 𝜇 is the mean and 𝜎 2 is the variance.
√2𝜋𝜎
Gaussian signals are important in communication systems because they simplify
analysis and enable the use of efficient signal processing techniques. Their behavior
 is fully characterized by mean and variance, allowing for easy computation of
probabilities and performance metrics. Gaussianity is often assumed for noise in
channels, guiding the design of robust modulation and coding schemes. It also aids
in filter and equalizer design, as many techniques are optimized for Gaussian inputs.
Engineers use statistical tests like the Shapiro-Wilk test to assess Gaussianity,
making it a crucial factor in communication system design and performance
analysis.
9. SIGNAL TO NOISE RATIO (SNR)
SNR is a measure of the strength of a signal relative to the background noise in a
communication system. It is typically expressed in decibels (dB) and is calculated
as
𝑃
SNR = 10𝑙𝑜𝑔10 ( 𝑃𝑠𝑖𝑔𝑛𝑎𝑙 )
𝑛𝑜𝑖𝑠𝑒

Where 𝑃𝑠𝑖𝑔𝑛𝑎𝑙 is the power of the signal and 𝑃𝑛𝑜𝑖𝑠𝑒 is the power of the noise

Signal-to-noise ratio (SNR) is a key metric in communication systems, with higher

values
indicating clearer, more reliable signals and lower values suggesting greater noise
interference. It directly affects bit error rate (BER), transmission quality, and
channel capacity, guiding decisions on modulation, coding, and error correction.
Engineers strive to maximize SNR through optimized transmission, quality
components, and by mitigating environmental factors like distance and interference
as described by Shannon’s theory C = 𝐵𝑙𝑜𝑔2 (1 + SNR) where C is the channel
capacity and B is the bandwidth.
10. BANDWIDTH
Bandwidth is the range of frequencies a signal occupies, measured in hertz, and
directly affects the data rate and capacity of communication systems. For a signal
with a frequency range from 𝑓1 to 𝑓2 , the bandwidth B is given by B = 𝑓1 - 𝑓2 .
Understanding bandwidth is essential in digital communications, as it governs
efficient spectrum use and limits the maximum data transmission rate as described
by Nyquist formula C = 2𝐵𝑙𝑜𝑔2 (M) where C is the channel capacity, B is the
 bandwidth and M is the number of discrete signal levels. Engineers design
communication systems with specific bandwidth requirements based on data type
and desired performance. The choice of modulation scheme significantly affects
the amount of bandwidth needed for effective transmission. Bandwidth allocation
must also account for regulatory constraints and potential interference from
adjacent channels. Overall, bandwidth is a key factor that influences system
performance, including latency, error rates, and transmission efficiency

C. Draw the modulation and demodulation diagram of AM, DSBSC, SSB and
VSB, respectively. Please draw necessary figures to show how to work in each
part of diagram on frequency domain.

AM Diagram

DSBSC Diagram

SSB Diagram
 VSB Diagram

D. Draw the modulation and demodulation diagram of NBFM and WBFM,

respectively.

NBFM Diagram

WBFM Diagran
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