Signal Processing — Foundation to Telecom Applications

Every phone call you make follows a fixed chain: your voice is captured as an electrical signal → converted to digital form → transmitted → reconstructed at the other end. Signal processing is the science behind each step of that chain. This page covers the concepts you are most likely to encounter in the BSNL SET exam.

A complete telecommunication system: voice → electrical signal → sampled and digitized → transmitted → reconstructed at the receiver.

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1. What is a Signal?

A signal is any quantity (voltage, current, light intensity) that varies with time to carry information.

Type	What it means	Example
Analog (Continuous)	Varies smoothly at every instant	Human voice, temperature
Digital (Discrete)	Takes only specific values (0 or 1)	Data on a computer network
Periodic	Repeats after a fixed time interval (period T)	Sine wave, clock signal
Aperiodic	Does not repeat	A single voice burst
Deterministic	Fully predictable from a formula	Sine wave carrier
Random (Stochastic)	Cannot be predicted exactly	Thermal noise

Key definitions:

• Frequency (f): Number of cycles per second, measured in Hertz (Hz). $f = 1/T$
• Amplitude: Peak value (strength) of the signal
• Phase: The starting position of a periodic signal in its cycle
• Bandwidth: The range of frequencies a signal occupies ($f_{high} - f_{low}$)

Telecom fact: A telephone voice channel occupies 300 Hz to 3400 Hz, giving it a bandwidth of about 3.1 kHz.

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2. Analog vs Digital Signals

The telecom world shifted from analog to digital because digital signals are far more resistant to noise and easier to process, store, and transmit.

Property	Analog	Digital
Signal form	Continuous waveform	Discrete bits (0 and 1)
Noise impact	Accumulates along the route	Can be detected and corrected
Regeneration	Signal degrades with distance	Can be perfectly regenerated
Processing	Harder to manipulate	Easy to process with computers
Example systems	Old PSTN telephone, AM/FM radio	GSM, fiber optic, internet

Exam point: Digital signals are preferred in modern telecom because noise does not accumulate — repeaters regenerate the exact bit pattern, not a degraded waveform.

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3. Sampling Theorem (Nyquist Theorem)

To convert an analog signal to digital, we first sample it — we measure its value at regular time intervals.

Nyquist Sampling Theorem: To perfectly reconstruct a signal, the sampling frequency must be at least twice the highest frequency in the signal.

$f_s \geq 2 \times f_{max}$

This minimum value ($2 \times f_{max}$) is called the Nyquist rate.

Standard telephony example:

• Maximum voice frequency = 4,000 Hz
• Nyquist rate = $2 \times 4,000$ = 8,000 Hz (8 kHz)
• PSTN samples voice at exactly 8 kHz — 8,000 samples every second

Aliasing

If sampling is done at a rate lower than the Nyquist rate, high-frequency components appear as wrong lower frequencies in the digital signal. This distortion is called aliasing.

Prevention: An anti-aliasing filter (a low-pass filter) is placed before the sampler to remove all frequencies above $f_s/2$ before sampling. This is a mandatory step in any ADC (Analog-to-Digital Converter) design.

Exam point: Anti-aliasing filtering always happens before sampling, never after.

Sampling process, aliasing effect when sampling rate is too low, quantization levels, and the PCM encoding chain.

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4. Quantization

After sampling, each sample has a continuous amplitude value. Quantization rounds each sample to the nearest level from a fixed set of discrete levels.

• Number of quantization levels = $2^N$, where $N$ = number of bits per sample
• More bits → more levels → finer resolution → better quality

Bits (N)	Levels ($2^N$)	Typical use
8	256	Standard telephony (G.711)
16	65,536	CD-quality audio

Quantization noise: The small error introduced by rounding is called quantization noise. More bits means smaller rounding error and better Signal-to-Noise Ratio (SNR).

Rule of thumb (exam-relevant): Each additional bit of quantization improves SNR by approximately 6 dB. An 8-bit PCM system gives roughly 50 dB SNR.

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5. PCM — Pulse Code Modulation

PCM is the standard method for converting an analog voice signal into a digital bitstream. It is the foundation of all digital telephony.

PCM process (in order):

1. Low-Pass Filter — removes frequencies above 4 kHz (anti-aliasing)
2. Sampler — takes 8,000 samples per second
3. Quantizer — rounds each sample to one of 256 levels (8-bit)
4. Encoder — converts each level to an 8-bit binary code

Calculating the bitrate of a PCM voice channel:

$\text{Bit Rate} = f_s \times N = 8{,}000 \times 8 = 64{,}000 \text{ bps} = \textbf{64 kbps}$

This is why a standard digital voice channel is always 64 kbps. This is the building block of all E1/T1 multiplexed telephony systems.

Companding (G.711): To make better use of the 8 bits for quieter sounds, the quantization steps are made non-uniform — finer at low amplitudes and coarser at high amplitudes. Two international standards exist:

• A-law — used in Europe and India
• μ-law — used in North America and Japan

Exam point: India follows the A-law companding standard. The G.711 ITU standard defines 64 kbps PCM voice coding.

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6. Filtering

A filter is a circuit or algorithm that allows certain frequencies to pass while blocking others.

Filter type	What it passes	Typical use
Low-Pass Filter (LPF)	Frequencies below a cut-off	Anti-aliasing, voice channel limiting
High-Pass Filter (HPF)	Frequencies above a cut-off	Removing DC components
Band-Pass Filter (BPF)	A specific range of frequencies	Selecting a radio channel
Band-Stop / Notch Filter	Blocks a specific range	Removing 50 Hz power-line interference

FIR vs IIR

Digital filters are broadly classified into two types based on how they process previous outputs:

Property	FIR (Finite Impulse Response)	IIR (Infinite Impulse Response)
Feedback	No (feed-forward only)	Yes (uses past outputs)
Stability	Always stable	Can become unstable
Phase response	Linear phase (no distortion)	Non-linear phase
Efficiency	Needs more coefficients	More efficient (fewer coefficients)
Use case	When phase accuracy matters	When efficiency matters

Exam point: FIR filters have no feedback and are always stable. IIR filters use feedback and can be unstable.

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7. Frequency Domain — Key Concept

Every signal can be described either in the time domain (how it varies over time) or the frequency domain (which frequencies it contains and at what strength).

The Fourier Transform is the mathematical tool that converts a signal from time domain to frequency domain. You do not need to evaluate the transform formula for the exam — what matters is understanding what it tells you:

• A pure sine wave at 1000 Hz appears as a single spike at 1000 Hz in the frequency domain
• A voice signal contains many frequencies between 300 Hz and 3400 Hz
• Filtering in the frequency domain is equivalent to multiplying by a window that passes or blocks certain frequencies

Power Spectral Density (PSD): Describes how the power of a signal is spread across frequencies. Wider bandwidth = more frequencies = more PSD spread.

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8. Noise and SNR

Noise is any unwanted signal that interferes with the desired signal.

Signal-to-Noise Ratio (SNR) measures how strong the signal is compared to the noise:

$SNR_{dB} = 10 \log_{10}\left(\frac{\text{Signal Power}}{\text{Noise Power}}\right)$

Higher SNR = better quality. A value of 0 dB means signal and noise are equally strong; 40 dB means the signal is 10,000 times stronger than the noise.

Common noise types in telecom:

• Thermal noise — caused by random electron movement in conductors; unavoidable
• Quantization noise — from rounding during digitization
• Crosstalk — interference from a neighboring channel
• Impulse noise — sudden spikes from switching equipment or lightning

Exam point: Thermal noise (also called Johnson noise) is present in all conductors and cannot be eliminated — only minimized by design.

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9. Modulation — Why It Matters for Signal Processing

Modulation shifts a low-frequency baseband signal onto a higher-frequency carrier for transmission. This is done because low-frequency signals cannot travel long distances efficiently through air.

Type	What changes on the carrier	Key use
AM (Amplitude Modulation)	Amplitude of the carrier	AM radio, DSB/SSB links
FM (Frequency Modulation)	Frequency of the carrier	FM radio, VHF communications
PM (Phase Modulation)	Phase of the carrier	Basis of digital modulations (PSK)

Demodulation is the reverse — extracting the original baseband signal from the modulated carrier at the receiver.

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10. Multiplexing

Multiplexing lets multiple signals share a single transmission medium simultaneously.

Type	How it divides the medium	Telecom use
FDM (Frequency Division Multiplexing)	Different frequency bands	Analog telephone trunk lines, cable TV
TDM (Time Division Multiplexing)	Different time slots	E1/T1 digital lines, SONET/SDH
WDM (Wavelength Division Multiplexing)	Different wavelengths (colors) of light	Fiber optic networks
CDM (Code Division Multiplexing)	Different spreading codes	CDMA mobile networks

TDM and PCM together: An E1 line carries 32 TDM channels, each a 64 kbps PCM voice stream. Total bitrate = $32 \times 64 \text{ kbps} = 2.048 \text{ Mbps}$.

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11. Quick Revision — Key Facts for the Exam

Topic	Remember this
Nyquist / Sampling theorem	$f_s \geq 2 f_{max}$
Voice sampling rate in PSTN	8,000 samples/sec (8 kHz)
PCM voice bitrate	64 kbps (8000 × 8 bits)
Quantization levels for N bits	$2^N$
SNR improvement per bit	~6 dB per bit added
Companding standard in India	A-law (G.711)
Anti-aliasing filter is placed	Before the sampler
FIR filter characteristic	No feedback, always stable, linear phase
IIR filter characteristic	Has feedback, efficient, can be unstable
Aliasing cause	Sampling below Nyquist rate
E1 line bitrate	2.048 Mbps (32 channels × 64 kbps)
Thermal noise	Present in all conductors; unavoidable

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12. Practice Questions

1. What is the minimum sampling rate required for a 4 kHz audio signal?
2. In PCM telephony, how many samples are taken per second?
3. How many quantization levels does an 8-bit PCM system use?
4. What is the bitrate of one standard PCM voice channel?
5. Which companding standard is used in India?
6. What is the purpose of the anti-aliasing filter in a PCM system?
7. What distortion occurs when a signal is under-sampled?
8. Which type of digital filter always has linear phase response?
9. A signal sampled at 8 kHz with 8 bits per sample — what is its bitrate?
10. What does FDM stand for, and how does it allow multiple users to share a medium?
11. Name the four steps in the PCM process, in order.
12. How does increasing the number of quantization bits affect signal quality?

1. 8 kHz (Nyquist rate = 2 × 4,000 Hz).
2. 8,000 samples per second.
3. $2^8 = 256$ levels.
4. 64 kbps (8,000 samples/sec × 8 bits/sample).
5. A-law (G.711 standard).
6. It removes frequencies above $f_s/2$ to prevent aliasing before the signal is sampled.
7. Aliasing — high-frequency components appear as incorrect lower frequencies.
8. FIR (Finite Impulse Response) filters.
9. 8,000 × 8 = 64,000 bps = 64 kbps.
10. Frequency Division Multiplexing — each user is assigned a different frequency band so signals do not interfere.
11. Low-Pass Filter → Sampler → Quantizer → Encoder.
12. More bits mean more quantization levels, finer amplitude resolution, less quantization noise, and better SNR (approximately +6 dB per additional bit).