Signals, Systems and Communication Experiments

Introduction

Signals and systems form the analytical foundation of electrical engineering. A signal represents information about a physical phenomenon, while a system transforms an input signal into an output. Communication engineering applies these ideas to convey information reliably through a channel in the presence of bandwidth limitations, distortion and noise.

This laboratory chapter develops practical competence in observing signals, measuring system behavior, moving between time and frequency domains, sampling and reconstructing signals, implementing filters, studying analog modulation and demodulation, and evaluating a basic digital communication link. The experiments are intended for laboratory instruments such as an oscilloscope, function generator, spectrum analyzer or oscilloscope FFT, digital multimeter, communication trainer, and computer-based data-acquisition software.

The laboratory should be conducted with disciplined measurement practice. Instrument grounds must be connected correctly, signal amplitudes must be specified consistently, and sampling rates must be sufficiently high for the signal under test. Results should include measured waveforms, numerical values, uncertainty or tolerance where appropriate, and a comparison between theoretical and experimental behavior.

Learning Objectives

On completing this chapter, a student should be able to:

1. Distinguish continuous-time, discrete-time, periodic, aperiodic, deterministic and random signals.
2. Measure amplitude, frequency, period, phase and power from time-domain waveforms.
3. Interpret the Fourier-series and Fourier-transform descriptions of signals.
4. Determine suitable sampling rates and explain aliasing.
5. Measure the magnitude and phase response of a filter.
6. Explain amplitude modulation, demodulation and modulation index.
7. Describe the effects of additive noise, channel attenuation and bandwidth limitation.
8. Implement a basic digital communication experiment and estimate bit-error performance.
9. Compare theoretical predictions with laboratory measurements and identify sources of error.

1. Signals and Systems: Laboratory Foundations

A continuous-time signal is written as x(t), where t is time in seconds. A discrete-time signal is written as x[n], where n is an integer sample index. Important classifications are summarized below.

For a periodic signal, the fundamental frequency is

where f₀ is in hertz and T₀ is in seconds. A sinusoid may be written as

where A is peak amplitude, f₀ is frequency, and φ is phase in radians. For a sinusoid of peak amplitude A across resistance R, the root-mean-square value and average power are

These expressions assume a resistive load and a sinusoidal waveform. They must not be applied to an arbitrary waveform without first determining its RMS value.

A system may be characterized as linear if it satisfies superposition, and time invariant if delaying the input only delays the output by the same amount. A linear time-invariant system is described by its impulse response h(t), with output

The convolution equation assumes an LTI system and suitable signal integrability. In the frequency domain, convolution becomes multiplication:

Here X(f), Y(f), and H(f) are the Fourier transforms of the input, output and impulse response. The transfer function H(f) describes amplitude scaling and phase shift as functions of frequency.

Laboratory precautions

Use a common reference ground only where permitted by the instrument design. Do not connect two grounded oscilloscope clips to points at different potentials. Confirm the function-generator output setting: some generators specify amplitude for a high-impedance load, whereas others assume a 50 Ω termination. Begin with a small amplitude and increase it only after confirming that no input is overloaded.

2. Experiment 1: Time-Domain Signal Measurement

Aim

To generate and measure sinusoidal, square and triangular signals and compare measured quantities with theoretical values.

Apparatus

Use a function generator, oscilloscope and, if required, a resistive load and frequency counter.

Procedure

1. Connect the generator to an oscilloscope channel using a suitable coaxial cable.
2. Generate a 1 kHz sine wave with a specified peak-to-peak amplitude.
3. Measure the period using horizontal cursors and calculate frequency using f = 1/T.
4. Measure peak-to-peak voltage and calculate peak and RMS values.
5. Repeat for square and triangular waves.
6. Observe the effects of DC offset and oscilloscope AC/DC coupling.
7. Record the measured waveform, frequency, amplitude and offset.

Worked example

A sinusoidal waveform has a measured peak-to-peak voltage of 4.00 V and frequency 2.00 kHz. Its peak amplitude is 2.00 V, and

The period is

Across a 100 Ω resistor, the average power is

The measured value may differ because of generator tolerance, oscilloscope resolution, probe attenuation and loading.

For a general periodic waveform, RMS voltage is defined by

This definition is valid for any periodic waveform. For example, an ideal symmetrical square wave with levels +A and −A has V_rms = A, whereas its average value is zero.

3. Experiment 2: Fourier Analysis and Frequency-Domain Interpretation

The Fourier series represents a periodic signal as a sum of harmonically related sinusoids. For a signal with period T₀ and fundamental angular frequency ω₀ = 2π/T₀,

The coefficients are

and

The Fourier transform extends frequency-domain analysis to aperiodic signals:

The inverse transform is

The transform convention above uses frequency f in hertz. Spectral magnitude is commonly displayed as |X(f)|, while angle X(f) gives phase.

Experimental procedure

Generate a square wave and display its FFT. Identify the fundamental and harmonics. A symmetrical square wave contains odd harmonics at 3f₀, 5f₀, 7f₀ and so on, with amplitudes decreasing approximately as 1/n for the nth odd harmonic under ideal conditions. Rise-time limitation, bandwidth, sampling settings and windowing affect the displayed spectrum.

For a finite record, the FFT assumes periodic repetition of the captured record. If the record does not contain an integer number of cycles, spectral leakage occurs. A window function reduces leakage but broadens spectral peaks. Therefore, frequency resolution and leakage are separate considerations: increasing record duration improves frequency resolution, while window selection affects leakage and peak shape.

4. Experiment 3: Sampling and Aliasing

Sampling converts x(t) into a sequence x[n] = x(nTₛ), where Tₛ is the sampling interval and fₛ = 1/Tₛ is the sampling frequency. For a band-limited signal with highest frequency B, ideal uniform sampling permits perfect reconstruction when

The quantity 2B is the Nyquist rate. This condition assumes ideal band limitation and an ideal reconstruction filter. In practical systems, the sampling frequency is selected above the theoretical minimum to allow for filter transition bands.

If a sinusoid of frequency f is sampled at fₛ, frequencies separated by integer multiples of fₛ produce the same sample sequence. An observed alias frequency may be calculated as the frequency folded into the interval from 0 to fₛ/2. For example, sampling a 7 kHz sine wave at 10 kHz produces an alias at

Procedure

1. Generate a sine wave whose frequency can be varied.
2. Acquire it at a known sampling rate.
3. Increase the input frequency beyond half the sampling rate.
4. Observe the apparent reduction or reversal in measured frequency.
5. Insert or enable an anti-aliasing low-pass filter and repeat.

An anti-aliasing filter must attenuate frequency components above fₛ/2 before sampling. Once aliasing has occurred, digital filtering after sampling generally cannot identify and remove the original out-of-band component because information has been irreversibly superimposed.

5. Experiment 4: Filter Frequency Response

A filter selectively passes or attenuates frequency components. For an LTI filter, the frequency response is

The magnitude in decibels is

The 20 factor applies to an amplitude or voltage ratio. A power ratio uses 10 log₁₀(P₂/P₁). The phase response is ∠H(f), usually reported in degrees or radians.

For a first-order RC low-pass filter with resistor R and capacitor C,

where ω = 2πf. Its cutoff frequency, defined as the frequency at which the magnitude is 1/√2 of the low-frequency magnitude, is

At f = f_c, the magnitude is −3.01 dB and the phase is −45° for the low-pass configuration. These statements assume ideal components and measurement at the capacitor output.

Worked example

For R = 10 kΩ and C = 10 nF,

At 15.9 kHz, approximately ten times the cutoff frequency, the magnitude is

Thus the attenuation is approximately −20.0 dB. Measure gain and phase at several frequencies and plot the Bode response. The experimental cutoff may differ because of component tolerances, generator output resistance, oscilloscope input impedance and parasitic capacitance.

6. Experiment 5: Amplitude Modulation and Demodulation

In amplitude modulation (AM), the amplitude of a carrier is varied according to a message signal. For a single-tone message,

where A_c is carrier amplitude, f_c is carrier frequency, f_m is message frequency, and μ is the modulation index. For conventional AM without overmodulation, 0 ≤ μ ≤ 1. Expanding the expression shows components at f_c and f_c ± f_m. The transmission bandwidth for a single-tone AM signal is 2f_m; for a message with highest frequency B, it is 2B.

The modulation index can be estimated from an oscilloscope envelope using maximum and minimum envelope amplitudes:

This assumes a correctly observed AM envelope and a carrier with no severe distortion. If μ > 1, envelope crossings occur and a simple envelope detector produces distortion.

An envelope detector uses a diode, capacitor and load resistor. The RC time constant must be long compared with the carrier period but short enough to follow the message variation. If it is too short, carrier ripple appears; if too long, diagonal clipping occurs.

Procedure

Generate a carrier, apply a low-frequency message, and adjust μ. Observe the time waveform and FFT. Demodulate using an envelope detector or a software detector. Compare the recovered message with the original and record distortion as μ is increased.

7. Experiment 6: Noise and Channel Behavior

Noise is an unwanted random signal. A commonly used model is additive white Gaussian noise (AWGN), in which the received signal is

where n(t) is modeled as zero-mean Gaussian noise with approximately constant power spectral density over the channel bandwidth. This is a model, not a statement that all physical noise is Gaussian or white.

Signal-to-noise ratio is

and in decibels,

For equal reference impedance, voltage measurements may use

Channel attenuation in decibels is similarly expressed as 20 log₁₀(V_out/V_in) for equal impedances. A negative value indicates attenuation.

A practical channel may exhibit attenuation, phase shift, limited bandwidth, multipath effects and frequency-selective fading. A filter can improve received quality by rejecting out-of-band noise, but it cannot restore information already lost through excessive attenuation or distortion.

Procedure

Transmit a sinusoid or AM signal through a variable-gain channel. Measure output amplitude and phase at several frequencies. Add controlled noise and observe the time waveform, spectrum and demodulated output. Record the minimum SNR at which the message remains intelligible or meets a specified measurement criterion.

8. Experiment 7: Basic Digital Communication Performance

A binary digital link maps bits into waveforms, transmits them through a channel, and decides which bit was sent. In binary amplitude-shift keying, for example, one amplitude represents binary 1 and another represents binary 0. In binary phase-shift keying (BPSK), the two symbols differ by 180° phase.

The bit-error rate is

where N_e is the number of incorrectly detected bits and N_b is the total number of tested bits. BER is an estimate of error probability; its reliability depends on the number of observed bits. If no errors are observed in a small sample, the true BER is not necessarily zero.

For coherent BPSK in AWGN, the theoretical bit-error probability is

where E_b is energy per bit, N₀ is the one-sided noise power spectral density, and Q(·) is the tail probability of a standard normal random variable. This result assumes ideal synchronization, coherent detection, independent AWGN and a matched receiver. It should not be used without these assumptions.

Procedure

1. Generate a pseudorandom binary sequence.
2. Modulate it using the selected digital modulation method.
3. Pass it through a channel with adjustable attenuation and noise.
4. Synchronize and demodulate the received signal.
5. Compare transmitted and received bit sequences.
6. Calculate BER for several SNR settings and plot BER against SNR.

A log-scale BER plot is useful because communication performance often changes by orders of magnitude. Timing errors, carrier-frequency offset, intersymbol interference and incorrect threshold selection may cause errors even when the nominal noise level is low.

9. Integrated Illustrative Case: Low-Power Sensor Link

Background

A laboratory team is designing a short-range link for a temperature sensor. The sensor produces a slowly varying analog voltage, but the system must transmit measurements through a noisy, bandwidth-limited cable.

Problem

Initial measurements show that direct transmission is affected by high-frequency interference. Sampling at 2 kHz produces occasional false rapid changes in the reconstructed temperature, suggesting aliasing. The communication receiver also reports errors when the cable is operated near a motor drive.

Analysis

The sensor bandwidth is estimated as 300 Hz. The minimum theoretical sampling rate is therefore greater than 600 samples/s, but the team selects 2 kHz to provide transition-band margin. An anti-aliasing low-pass filter is inserted before the ADC. A spectrum measurement identifies interference near 1.2 kHz, which would fold into the measurement band if sampling were performed without adequate attenuation.

The team then compares analog transmission with a binary digital link. Digital transmission is less sensitive to small amplitude variations but requires synchronization and produces abrupt performance degradation when the SNR becomes too low.

Approach and outcome

A second-order low-pass anti-aliasing filter is selected, followed by sampling at 2 kHz. The digital link uses a pseudorandom test sequence and records BER at several noise levels. Shielding and correct cable termination reduce conducted interference. Measurements confirm that the filtered system has a stable reconstructed output, while the BER curve identifies the minimum operating SNR for the required reliability.

Learning points

The case demonstrates that sampling, filtering, channel analysis and communication performance must be designed together. A higher sampling rate alone does not correct poor analog filtering, and a digital link does not eliminate the need for careful grounding, bandwidth control and synchronization.

10. Measurement Uncertainty, Safety and Professional Practice

Report nominal instrument accuracy, resolution and repeatability where relevant. Repeat measurements rather than relying on a single cursor reading. Distinguish systematic error, such as probe attenuation, from random variation, such as noise-trigger jitter. Use consistent definitions of peak, peak-to-peak, RMS and average values.

Electrical safety is essential. Never exceed instrument input ratings. Discharge capacitors before modifying a circuit. Use isolation only when appropriate and approved; an isolation transformer is not a substitute for safe measurement procedure. Protect communication data when experiments involve recorded speech or identifiable personal information. Laboratory data should not be altered to match theory; discrepancies are evidence to be analyzed.

Modern instruments often use software, FFT algorithms and automated measurements. These tools improve efficiency but do not remove the need to understand sampling, windowing, calibration and signal paths. AI-assisted measurement or coding may help generate plots or test scripts, but results must be independently verified against known signals and documented procedures.

Chapter Summary

Signals describe information, and systems transform signals. Time-domain measurements reveal amplitude, period, phase and waveform shape, while Fourier analysis reveals spectral components and system bandwidth. LTI systems are analyzed through convolution in time and multiplication by the transfer function in frequency. Sampling requires a rate greater than twice the highest signal frequency under ideal band-limited assumptions; anti-aliasing filtering is therefore essential. Filter experiments connect transfer functions with measured gain and phase. AM experiments demonstrate carrier, sidebands, modulation index and envelope detection. Noise and channel experiments introduce SNR, attenuation and bandwidth limitations. Digital communication experiments use BER to quantify reliability, but theoretical results require stated assumptions. Sound laboratory practice includes safe connections, calibration awareness, appropriate sampling, repeatability and honest interpretation of discrepancies.

Key Terms

Signal: A physical quantity that conveys information.

System: A process or device that maps an input signal to an output.

LTI system: A system that is both linear and time invariant.

Convolution: The time-domain operation describing an LTI system response.

Fourier series: Representation of a periodic signal as harmonically related sinusoids.

Fourier transform: Frequency-domain representation of an aperiodic or general signal.

Spectrum: Distribution of signal magnitude or power with frequency.

Sampling: Conversion of a continuous-time signal into discrete-time samples.

Nyquist rate: Twice the highest frequency of an ideally band-limited signal.

Aliasing: Misidentification of frequency caused by insufficient sampling.

Anti-aliasing filter: Analog filter that suppresses frequencies above the usable sampling band.

Bandwidth: Frequency range occupied or passed by a signal or system.

Modulation: Variation of a carrier parameter to convey information.

Modulation index: Normalized measure of message amplitude in AM.

Demodulation: Recovery of information from a modulated carrier.

AWGN: Additive white Gaussian noise model.

SNR: Ratio of signal power to noise power.

BER: Ratio of incorrectly detected bits to total tested bits.

Intersymbol interference: Distortion in which neighboring symbols overlap.

Spectral leakage: Spreading of FFT energy caused by finite, non-periodic records.

Review Questions

Short Answer Questions

1. Define a continuous-time signal and a discrete-time signal.
2. What is the physical meaning of the cutoff frequency of a first-order low-pass filter?
3. Why is an anti-aliasing filter used before an analog-to-digital converter?
4. Define modulation index for conventional AM.
5. What does BER measure in a digital communication system?

Long Answer Questions

1. Explain the relationship among impulse response, convolution and transfer function for an LTI system.
2. Describe a complete laboratory procedure for measuring the frequency response of an RC low-pass filter.
3. Explain Fourier leakage and discuss how record length and window selection affect an FFT.
4. Derive the spectral components of a single-tone AM signal and state its bandwidth.
5. Discuss the effects of attenuation, noise, limited bandwidth and synchronization errors on a communication link.

Analytical / Application Questions

1. A 4 kHz sinusoid is sampled at 6 kHz. Determine its apparent alias frequency and explain how an anti-aliasing filter would prevent the problem.
2. An RC low-pass filter has R = 4.7 kΩ and C = 22 nF. Calculate its cutoff frequency and state the expected magnitude and phase at cutoff.
3. A digital receiver tests 200,000 bits and detects 18 errors. Calculate the measured BER and explain why the result is only an estimate of the underlying error probability.

Multiple Choice Questions with Answers

Question 1

A 2 kHz sinusoid has what period?

A. 0.05 ms

B. 0.5 ms

C. 2 ms

D. 500 ms

Correct Answer: B. 0.5 ms

Question 2

For a 1 kHz sinusoid sampled at 1.5 kHz, which phenomenon is most likely?

A. Aliasing

B. Perfect reconstruction

C. DC elimination

D. Amplitude modulation

Correct Answer: A. Aliasing

Question 3

The cutoff frequency of an ideal first-order RC low-pass filter is defined at which magnitude?

A. The input magnitude

B. Twice the input magnitude

C. 1/√2 of the low-frequency magnitude

D. Zero magnitude

Correct Answer: C. 1/√2 of the low-frequency magnitude

Question 4

A symmetrical ideal square wave contains which harmonics?

A. Only even harmonics

B. Only odd harmonics

C. No harmonics

D. Only subharmonics

Correct Answer: B. Only odd harmonics

Question 5

For conventional AM without envelope distortion, the modulation index should generally satisfy:

A. μ < 0 only

B. μ = 10

C. μ ≥ 5

D. 0 ≤ μ ≤ 1

Correct Answer: D. 0 ≤ μ ≤ 1

Question 6

If signal power is four times noise power, the SNR in decibels is approximately:

A. −6.02 dB

B. 2 dB

C. 6.02 dB

D. 40 dB

Correct Answer: C. 6.02 dB

Question 7

A system tests 10,000 bits and receives 25 incorrectly. Its measured BER is:

A. 0.00025

B. 0.0025

C. 0.025

D. 2.5

Correct Answer: B. 0.0025

Question 8

For an RC low-pass filter with R = 10 kΩ and C = 10 nF, the cutoff frequency is closest to:

A. 15.9 Hz

B. 159 Hz

C. 1.59 kHz

D. 15.9 kHz

Correct Answer: C. 1.59 kHz

Question 9

Which operation changes convolution in time into multiplication in frequency for an LTI system?

A. Fourier transformation

B. Differentiation only

C. Quantization

D. Sampling without filtering

Correct Answer: A. Fourier transformation

Question 10

In an FFT of a finite record containing a non-integer number of cycles, the spreading of energy across frequency bins is called:

A. Quantization

B. Spectral leakage

C. Demodulation

D. Carrier suppression

Correct Answer: B. Spectral leakage