• Distinguish analog and digital data and signals, periodic and aperiodic
• Describe a sine wave by amplitude, frequency, phase and wavelength
• Read a signal in the time domain and the frequency domain
• Explain composite signals and calculate bandwidth
• Relate bit rate, bit length and the bandwidth a digital signal needs

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Lecture 2: Data and Signals I

University of Barishal

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• Analog data is continuous: voice, temperature, light level
• Digital data is discrete: text, numbers, files stored as bits
• A signal is the electromagnetic form of data that can travel through a medium
• Both data and signals can be analog or digital, giving four combinations (right)

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Lecture 2: Data and Signals I

University of Barishal

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• An analog signal changes smoothly and can take infinitely many values in a range
• A digital signal has a limited number of values, often just two
• Flat lines: the value is held; vertical lines: an instant jump to a new level
• Example: a microphone's output voltage is analog; the data line of a USB cable is digital

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Lecture 2: Data and Signals I

University of Barishal

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• Periodic: completes a pattern in a time called the period, then repeats it exactly
• Aperiodic (nonperiodic): changes without any repeating pattern
• Data communication typically uses periodic analog signals as carriers and aperiodic digital signals for data
• Examples: Bangladesh mains electricity (periodic) and your speech (aperiodic)

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Lecture 2: Data and Signals I

University of Barishal

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Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

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• Equation: s(t) = A sin(2πft + φ)
• It cannot be split into simpler signals; every other signal is built from sine waves
• Three numbers describe it completely: peak amplitude A, frequency f (or period T) and phase φ
• Animation: the circle's radius is A, its turning speed is f, and its starting angle is φ

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Lecture 2: Data and Signals I

University of Barishal

7 / 33

• Peak amplitude: the largest absolute value the signal reaches
• Measured in volts, amperes or watts, depending on the signal
• Changing amplitude changes only the height, not the timing
• A 1.5 V battery gives a constant voltage: amplitude 1.5 V, frequency 0 Hz

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

8 / 33

• Period T: the time for one complete cycle, in seconds
• Frequency f: the number of cycles per second, in hertz (Hz)
• They are inverses: f = 1 / T and T = 1 / f
• Fast change means high frequency; no change means 0 Hz; an instant jump means infinite frequency

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

9 / 33

• Problem 1: a sine wave completes one cycle in 100 ms. Find f.
• 100 ms = 0.1 s, so f = 1 / 0.1 = 10 Hz
• Problem 2: what is the period of a 2.4 GHz Wi-Fi signal?
• T = 1 / (2.4 × 10⁹) = 4.17 × 10⁻¹⁰ s = 0.417 ns

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

10 / 33

• Phase: the position of the waveform relative to time zero
• Measured in degrees or radians: one full cycle = 360° = 2π rad
• A shift of 1/4 cycle = 90° = π/2 rad; 1/2 cycle = 180° = π rad
• Phase changes where the wave starts, not its height or frequency
• Example: a shift of 1/6 cycle = 60° = π/3 rad

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Lecture 2: Data and Signals I

University of Barishal

11 / 33

• Wavelength λ: how far the signal travels during one period
• λ = propagation speed × T = c / f
• Speed in air or vacuum c ≈ 3 × 10⁸ m/s; in copper or fibre about 2 × 10⁸ m/s
• Higher frequency means shorter wavelength

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

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Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

13 / 33

• Time-domain plot: amplitude against time, what an oscilloscope shows
• Frequency-domain plot: peak amplitude against frequency, what a spectrum analyser shows
• A sine wave becomes a single spike: its height is A, its position is f
• The frequency view is compact and shows directly how much bandwidth a signal needs

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

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Try it: one sine wave, two views

Interactive

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

15 / 33

• Fourier: any signal can be written as a sum of sine waves with different A, f and φ
• A periodic composite signal has discrete spikes (Fourier series)
• An aperiodic composite signal has a continuous spectrum (Fourier transform)
• Analogy: a prism splits white light into its colours; Fourier splits a signal into its frequencies

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

16 / 33

• A square wave = f + 3f + 5f + … : the odd harmonics of the fundamental f = 1/T
• Harmonic n has amplitude proportional to 1/n, so higher harmonics are weaker
• Each extra harmonic sharpens the edges (animation, right)
• A perfect square wave needs infinitely many harmonics, so infinite bandwidth

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

17 / 33

Try it: build a square wave from sines

Interactive

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Lecture 2: Data and Signals I

University of Barishal

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• 1 · Sine wave: change A, f and φ; watch both domains
• 2 · Square wave: add harmonics until the edges are sharp
• 3 · Bits through a channel: shrink the bandwidth until bits break
• 4 · Calculators: period ↔ frequency, wavelength, bit length

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Lecture 2: Data and Signals I

University of Barishal

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• The spectrum of a signal is the set of all frequencies it contains
• Bandwidth B = highest frequency − lowest frequency, in hertz
• Signals have a bandwidth, and so do channels: the range of frequencies a channel passes
• Example: a telephone voice channel passes about 300 to 3,400 Hz, so B ≈ 3,100 Hz

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

20 / 33

• (a) A periodic signal contains 100, 300, 500, 700 and 900 Hz. B = 900 − 100 = 800 Hz
• (b) B = 20 Hz and the highest frequency is 60 Hz. Lowest = 60 − 20 = 40 Hz
• (c) An AM station uses a 10 kHz band centred on 1,000 kHz: it spans 995 to 1,005 kHz

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

21 / 33

• Engineers use “bandwidth” for two related but different quantities
• Hertz describes the width of the frequency range
• Bits per second describes the data capacity
• More hertz allows more bits per second, but how many also depends on signal levels and noise (Nyquist and Shannon, Lecture 3)

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Lecture 2: Data and Signals I

University of Barishal

22 / 33

• A digital signal uses a fixed number of voltage levels L
• Each level carries log₂ L bits: 2 levels → 1 bit, 4 → 2 bits, 8 → 3 bits
• More levels send more bits in the same time
• But levels sit closer together, so noise confuses them more easily

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Lecture 2: Data and Signals I

University of Barishal

23 / 33

• Bit rate: bits sent per second (bps); for digital signals it replaces frequency
• Bit interval = 1 / bit rate: the time one bit lasts (replaces period)
• Bit length = propagation speed × bit interval: the distance one bit occupies (replaces wavelength)
• Units: kbps = 10³, Mbps = 10⁶, Gbps = 10⁹ bits per second

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

24 / 33

• Text: 100 pages/s, 24 lines × 80 characters per page, 8 bits per character
• HD video: 1920 × 1080 pixels, 30 frames/s, 24 bits per pixel
• Voice (from Lecture 1): 8,000 samples/s, 8 bits per sample
• Method: multiply everything sent per second down to bits

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

25 / 33

• A digital signal is a sum of many harmonics, so its exact spectrum is infinitely wide
• We only need to send its significant spectrum: enough harmonics to read the bits
• If a channel's bandwidth is too narrow, harmonics are lost and the shape blurs
• Too much blurring and the receiver can no longer tell 0 from 1

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

26 / 33

• Worst case: alternating bits 0101… form a square wave with f = N/2 (one cycle per 2 bits)
• Keeping only the first harmonic needs at least B ≈ N/2
• Keeping more harmonics gives cleaner edges but needs 3N/2, 5N/2 …
• This applies to baseband transmission over a low-pass channel

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

27 / 33

Try it: squeeze bits through a channel

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Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

28 / 33

• Baseband: send the digital signal directly; needs a channel that passes frequencies from 0 Hz upwards
• Broadband (band-pass): modulate the signal onto a carrier so it fits a band away from 0 Hz
• Broadband lets many users share one medium, each in its own band
• Coming up: line coding for baseband (Lecture 6), modulation for broadband (Lecture 7)

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

29 / 33

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

30 / 33

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

32 / 33

• Data travels as signals; both can be analog or digital
• A sine wave is described by amplitude, frequency (f = 1/T), phase and wavelength (λ = c/f)
• Any signal is a sum of sine waves; its spectrum shows which ones
• Bandwidth = highest − lowest frequency; channels have bandwidth too
• Digital signals need bandwidth in proportion to bit rate: at least about N/2

Md Mahbub E Noor

Lecture 2: Data and Signals I

University of Barishal

33 / 33