Skip to main content

Why is Time-bandwidth Product (TBP) Important?


📘Theory 📏Example 🧮Filter Code 📈Gaussian Code 📚Further Reading

Time-Bandwidth Product (TBP)

The time-bandwidth product (TBP) is defined as:

TBP = Δf ⋅ Δt
  • Δf (Bandwidth): The frequency bandwidth of the signal, representing the range of frequencies over which the signal is spread.
  • Δt (Time duration): The duration for which the signal is significant, i.e., the time interval during which the signal is non-zero.

The TBP is a measure of the "spread" of the signal in both time and frequency domains. A higher TBP means the signal is both spread over a larger time period and occupies a wider frequency range.

To calculate the period of a signal with finite bandwidth, Heisenberg’s uncertainty principle plays a vital role where the time-bandwidth product indicates the processing gain of the signal.

We apply spread spectrum techniques in wireless communication for various reasons, such as interference resilience, security, robustness in multipath, etc. But in spread spectrum techniques, we compromise some bandwidth.

The time-bandwidth product for Gaussian-shaped pulses is 0.44 (approx.).

If the time-bandwidth product of a signal is >> 1, then the signal bandwidth (B) is much greater than what is required for transmitting the data rate (Rb). So, in this case, we are unable to utilize the whole available bandwidth. For this case, spectrum efficiency will be less.

To your knowledge, the product of the variance of time and variance of bandwidth for a Gaussian signal is 0.25, and for a triangular-shaped signal, it is 0.3.

Example: Raised Cosine Filter

Let’s assume we have designed a raised cosine filter with a roll-off factor of 0.25. The symbol rate for transmission is 100 symbols per second, and the number of samples per symbol is 10. Also, assume the filter span is 2, meaning the duration is up to 2 symbol times.

Bandwidth Calculation:

The bandwidth of the raised cosine filter is calculated as:

Bandwidth = (Symbol Rate × (1 + Roll-off Factor)) / 2
Bandwidth = (100 × (1 + 0.25)) / 2 = 62.5 Hz

Time Duration (Filter Span = 2):

Filter Duration = Filter Span × One Symbol Duration
Filter Duration = 2 × 0.01 = 0.02 seconds

Time-Bandwidth Product (TBP):

TBP = 0.02 × 62.5 = 1.25

Time Duration (Filter Span = 6):

If the filter span is 6, then the time-bandwidth product will be:

TBP = 0.06 × 62.5 = 3.75

Conclusion: The raised cosine filter reduces the effect of intersymbol interference (ISI) during signal transmission. Increasing the bandwidth helps mitigate ISI to a greater extent, but it also increases the time-bandwidth product, making the system less bandwidth-efficient.

Ready to Simulate?

Use the professional MATLAB scripts below to visualize the Time-Bandwidth Product in real-time.

View MATLAB Scripts ↓

MATLAB: Raised Cosine Filter TBP

MATLAB Script 
% The code is developed by SalimWireless.Com
clc;
clear;
close all;

% Parameters
beta = 0.25; % Roll-off factor
span = 2; % Filter span in symbols
sps = 10; % Samples per symbol
symbolRate = 1e2; % Symbol rate in Hz

% Generate the Raised Cosine Filter
rcFilter = rcosdesign(beta, span, sps, 'sqrt');

% Plot the Impulse Response
t = (-span/2 : 1/sps : span/2) * (1/symbolRate);
figure;
subplot(3,1,1);
plot(t, rcFilter, 'LineWidth', 1.5);
title('Raised Cosine Filter Impulse Response');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;

% Analyze Frequency Response
[H, F] = freqz(rcFilter, 1, 1024, sps * symbolRate);
subplot(3,1,2);
plot(F, abs(H), 'LineWidth', 1.5);
title('Raised Cosine Filter Frequency Response');
xlabel('Frequency (Hz)');
ylabel('Magnitude');
grid on;

% Time-Bandwidth Product Calculation
timeDuration = span * (1 / symbolRate); 
bandwidth = (1 + beta) * (symbolRate / 2); 
TBP = timeDuration * bandwidth; 

% Display Results
disp(['Time Duration (s): ', num2str(timeDuration)]);
disp(['Bandwidth (Hz): ', num2str(bandwidth)]);
disp(['Time-Bandwidth Product: ', num2str(TBP)]);

% Simulate Filtered Signal
numSymbols = 100;
data = randi([0 1], numSymbols, 1) * 2 - 1;
upsampledData = upsample(data, sps);
txSignal = conv(upsampledData, rcFilter, 'same');

subplot(3,1,3);
plot(txSignal(1:200), 'LineWidth', 1.5);
title('Filtered Transmitted Signal');
xlabel('Sample Index');
ylabel('Amplitude');
grid on;

Output Results

Time Duration (s): 0.02
Bandwidth (Hz): 62.5
Time-Bandwidth Product: 1.25

MATLAB: Gaussian Noise TBP

MATLAB Script 
% The code is developed by SalimWireless.Com
clc;
clear;
close all;

% Step 1: Generate Gaussian pulse
t = 0:0.01:1; % Time vector
sigma = 1; % Standard deviation
gaussian_pulse = exp(-t.^2 / (2 * sigma^2)); 

% Step 2: Calculate RMS time duration
power_signal = gaussian_pulse.^2;
rms_time = sqrt(sum(t.^2 .* power_signal) / sum(power_signal));

% Step 3: Calculate Frequency Bandwidth
Fs = 100; % Sampling frequency
N = length(gaussian_pulse);
f = (-N/2:N/2-1) * (Fs / N); % Frequency vector
G_f = fftshift(fft(gaussian_pulse)); % Fourier transform

power_spectrum = abs(G_f).^2;
rms_freq = sqrt(sum(f.^2 .* power_spectrum) / sum(power_spectrum));

% Step 4: Compute TBP
TBP_rms = rms_time * rms_freq;

% Display results
disp(['RMS Time Duration (Delta t): ', num2str(rms_time)]);
disp(['RMS Frequency Bandwidth (Delta f): ', num2str(rms_freq)]);
disp(['Time-Bandwidth Product (TBP): ', num2str(TBP_rms)]);

Output Results

RMS Time Duration (Delta t): 0.50383
RMS Frequency Bandwidth (Delta f): 0.98786
Time-Bandwidth Product (TBP): 0.49772

1. Simulator: Data Pulse (Raised Cosine)

This mimics how a single bit of data is shaped in modern wireless communication.

(Adjusts how "sharp" the filter is)

(How long the pulse lasts)

Calculated TBP: 1.25 Good Efficiency

2. Simulator: Gaussian Pulse (The Perfect Balance)

The Gaussian pulse is special because it achieves the minimum possible TBP. It is the "smoothest" possible signal.

(Widening in time automatically narrows frequency)

Calculated TBP: 0.44 Fundamental Minimum

3. The Math Behind the Simulators

The Time-Bandwidth Product (TBP) is like a "Space-Time" budget for signals. No matter how you design a signal, you cannot make it infinitely small in both time and frequency at once.

  • The Formula: TBP = Δt × Δf
  • The Limit: For any signal, TBP ≥ 0.5 (approx). You can never go below this limit. This is the Heisenberg Uncertainty Principle applied to signals.
  • The Trade-off:
    • If TBP ≈ 0.5 to 1.5: High Spectral Efficiency. Used in 5G, Wi-Fi, and Fiber Optics.
    • If TBP > 10: Spread Spectrum. Used in GPS and Radar to resist interference.

Real-world Applications of Different TBP

The choice of TBP is a strategic decision based on the application. It defines the "shape" of the energy in the time-frequency plane.

System Type Required TBP Primary Goal
Consumer Wireless (5G/Wi-Fi) $\approx 1.0$ High Spectral Efficiency; fitting max bits into narrow Hz.
GSM/GMSK (2G Mobile) $0.3$ (BT Product) Constant envelope for power-efficient amplifiers.
Radar Systems $> 10$ to $1000+$ Pulse Compression; High resolution with high energy.
Satellite Links High (>10) Robustness against deep space interference/fading.

Read More: about Time-Bandwidth Product and Pulse Shaping

GMSK Spectrum


📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

UGC NET Electronic Science Previous Year Question Papers with Solutions

Download Papers and Solutions Exam Pattern Preparation Tips FAQs More Home / Engineering & Other Exams / UGC NET 2026 PYQ 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2026 - Question Paper Download PDF June 202...
Read more

GMSK BER vs SNR (with Simulation)

  GMSK Performance: Why the BER vs. SNR Curve Matters If you’ve ever wondered why GMSK (Gaussian Minimum Shift Keying) became the backbone of the global GSM revolution, the answer lies in its unique balance of spectral efficiency and power performance. But there is no free lunch in engineering. To get that sleek, narrow spectrum, you have to pay a price in Bit Error Rate (BER) . In this guide, we break down the theoretical BER vs. SNR performance of GMSK, the "magic" of the BT product, and how to optimize your link budget for real-world conditions. What is GMSK and Why Should You Care? GMSK is a derivative of MSK (Minimum Shift Keying) that uses a Gaussian pre-modulation filter. This filter "smooths" the phase transitions, significantly reducing sideband power. The result? A constant envelope signal that allows you to use high-efficiency Class-C amplifiers without distorting the signal. This is the secr...
Read more

The output voltage V0 (in Volt) for the network given in the Figure is ___

  Find the Output Voltage V o Solve the given circuit using Kirchhoff's Current Law (KCL), Ohm's law, and the supernode method. 1. Given Circuit Parameters Current source 1 mA Resistors 1 kΩ Each resistor Voltage source 6 V Required V o 2. Understand the Circuit The circuit contains a 1 mA current source, four 1 kΩ resistors, and a 6 V source connected between the two midpoint nodes. The output voltage is measured between the top and bottom rails. Reference node Take the bottom rail as ground: 0 V . Let the top rail voltage be V T , and...
Read more

What is a Supernode?

  Circuit Analysis What Is a Supernode? A supernode is formed when an ideal voltage source connects two non-reference nodes. It allows us to apply Kirchhoff's Current Law (KCL) to the combined circuit boundary. 1. Understanding a Supernode In nodal analysis, a voltage source between two unknown-voltage nodes makes the current through that source difficult to express directly using Ohm's law. We combine the two nodes and the voltage source into one extended region called a supernode. Key idea Apply KCL around the outer boundary of the supernode, then use the voltage source to write an additional voltage equation. 2. Supernode Circuit Diagram The voltage source connects nodes V L an...
Read more

A binary ripple counter is designed to count (0)₁₀ to (64)₁₀. Which of the following is/are the number of flip-flops required ...

  Q. A binary ripple counter is designed to count  (0)₁₀ to (64)₁₀ . Which of the following is/are the number of flip-flops required to design the counter?   (A) 6 (B) 7 (C) 4 (D) 5 Solution A binary ripple counter must count from (0)₁₀ to (64)₁₀ , including both endpoints. Step 1: Calculate the total number of states Since both 0 and 64 are included, add 1 to the difference: N = Last value − First value + 1 N = 64 − 0 + 1 = 65 states The counter visits every integer from 0 through 64: 0, 1, 2, 3, ..., 63, 64 There are 65 distinct states in total. Step 2: Calculate the number of flip-flops Each flip-flop stores one bit. With n flip-flops, the counter can represent 2ⁿ distinct states. 2⁶ = 64 states (not enough) 2⁷ = 128 states (eno...
Read more

Parametric Down-Conversion of Photons

  Nonlinear Optics One incoming photon is converted into two lower-energy photons, typically in a nonlinear optical medium. Energy is conserved: E in = E 1 + E 2 E in ​ = E 1 ​ + E 2 ​ . Parametric Down-Conversion A nonlinear optical process in which one pump photon is converted into two lower-energy photons, known as the signal and idler photons. 1 Energy Conservation The energy of the pump photon equals the sum of the energies of the signal and idler photons. \[ \boxed{\hbar\omega_p=\hbar\omega_s+\hbar\omega_i} \] \[ \boxed{\omega_p=\omega_s+\omega_i} \] ωₚ Pump angular frequency ...
Read more

SRAM, DRAM, EPROM, MRAM, and DDRAM Explained

Computer Fundamentals Types of Memory and Their Technologies Understand the fundamental relationship between different types of computer memory and the technologies used to implement them. Fundamental Matching Memory type, technology, and key characteristics Memory Type Technology Key Characteristic Cache Memory SRAM Very fast; stores frequently accessed data Main Memory DRAM Holds programs and data currently in use Au...
Read more

The cutoff frequency (in GHz) for the dominant TE10 mode of an air-filled rectangular waveguide of ...

  Q. The cutoff frequency (in GHz) for the dominant TE10 mode of an air-filled rectangular waveguide of inner dimension 0.28 inch×0.14 inch is __. (rounded off to two decimal places). Given Information Waveguide dimensions = 0.28 inch × 0.14 inch Wider dimension (a) = 0.28 inch Smaller dimension (b) = 0.14 inch Medium = Air Dominant mode = TE 10 Speed of light (c) = 3 × 10 8 m/s Step 1: Cutoff Frequency Formula The cutoff frequency of a rectangular waveguide is: f c = (c / 2) √[(m/a) 2 + (n/b) 2 ] For the dominant TE 10 mode: m = 1, n = 0 f c = c / 2a Step 2: Convert Inch into Meter We know that 1 inch = 0.0254 meter. a = 0.28 × 0.0254 a = 0.007112 m ...
Read more