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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


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