Overview
Frequency-domain and time-domain analysis are fundamental methods for analyzing signals. They describe signal characteristics from different perspectives. A signal's properties can be represented in both the time domain and the frequency domain.
Frequency vs Time
Basic Methods for Signal Analysis
When discussing the relationship between frequency and time domains, start with the basic analysis methods for signals. Traditionally, analysis of wireless and wired communication signals is divided into three domains: time domain, frequency domain, and modulation domain. The modulation domain analyzes how frequency or phase varies with time.
Relationship among frequency domain, time domain, and amplitude
Frequency-domain Measurements
- Wide-frequency-range signal search
- Spurious signal testing
- Signal power parameters
- Occupied bandwidth of the signal
Time-domain Measurements
- Signal variation over time
Demodulation Measurements
- Signal modulation parameters
- Modulation accuracy
Demodulation measurements characterize the amplitude, phase, and frequency variations of modulated signals. This provides an alternative perspective to the traditional three domains and their corresponding instruments: oscilloscope, spectrum analyzer, and modulation-domain analyzer. The concept of demodulation measurement corresponds to vector signal analyzers.
Definitions: Time Domain and Frequency Domain
Time domain: analyzes how signal parameters change over time. The time domain represents the overall behavior of a signal along the time axis. In the time domain, all frequency components of a signal are summed and displayed. Oscilloscopes are typically used for time-domain observation.
Frequency domain: analyzes the frequency components contained in a signal and their amplitude and phase relationships. In the frequency domain, composite signals are separated into their frequency components and the level of each frequency is displayed. Spectrum analyzers are typically used for frequency-domain observation.
Because signals vary with time and also contain frequency and phase information, it is necessary to analyze the signal's frequency structure. Dynamic signals are transformed from the time domain to the frequency domain mainly using Fourier series and Fourier transforms.
Transforming Time-domain Functions to Frequency-domain Functions
In simple terms, a time-domain function has the independent variable t, i.e., y = f(t). The frequency-domain representation has the independent variable omega, i.e., Y = F(omega). These representations can be converted to each other. Time-domain functions become frequency-domain functions via Fourier or Laplace transforms.
Transformations between frequency-domain and time-domain analysis
Why Perform Frequency-domain Analysis? An Example with Square Waves
Oscilloscopes are commonly used to measure time-domain waveforms. Time-domain analysis shows amplitude, frequency, and phase variations in an intuitive way. However, time-domain observation alone may not reveal the cause of waveform differences.
What Is Frequency-domain Analysis?
Frequency-domain analysis examines a signal in frequency coordinates. A complete frequency-domain analysis identifies the frequency components present in the signal and the amplitude and phase relationships of each component. In other words, it analyzes the signal power spectrum and phase spectrum. When a waveform changes, its spectral characteristics change accordingly. Frequency-domain and time-domain analyses are basic methods for describing signal characteristics from different perspectives.
Frequency-domain analysis includes:
- Analyzing the frequency components in a signal and the frequency and power parameters of those components.
- Measuring signal power, occupied bandwidth, out-of-band spurious emissions, and ACPR.
History of Time-domain Reflectometry and Time-domain Analysis
Time domain reflectometry (TDR) was introduced in the early 1960s. It uses the same operating principle as radar: send an impulse into a cable or other device under test; when the impulse reaches the cable end or a fault point, part or all of the impulse is reflected back. TDR sends an impulse or step excitation into the device under test and observes the response in the time domain. In testing, a step generator and a wideband oscilloscope are used. The fast-rising edge from the step generator is sent into the transmission line under test, and incident and reflected voltage waveforms are observed with the wideband oscilloscope. By measuring the ratio of incident to reflected voltage, the impedance at the discontinuity can be calculated, and the location of that discontinuity can be determined from the propagation velocity along the transmission line. The nature of the impedance discontinuity, whether capacitive or inductive, can be identified from the response characteristics.
Although TDR oscilloscopes have been useful qualitative tools, there are limitations affecting measurement accuracy and effectiveness: a) the rise time of the TDR output step signal, since spatial resolution depends on the rise time; and b) not-ideal signal-to-noise ratio, caused by the structure of the oscilloscope wideband receiver.
In the 1970s, researchers showed that the relationship between frequency and time domains can be described using the Fourier transform.
A frequency-dependent network reflection coefficient can be converted by Fourier transform into a time-varying reflection coefficient, for example as a function of distance along a transmission line. This makes it possible to measure a device's response in the frequency domain and then compute the time-domain response by inverse Fourier transform.
Today, high-performance vector network analyzers (VNAs) have fast computation capabilities and offer unique measurement options. Error-corrected frequency-domain test data can be used to calculate a network's response to step or impulse excitation and display it as a time function. This provides time-domain reflection and transmission testing capabilities for networks with limited bandwidth. VNAs can perform more precise time-domain tests because they can identify unwanted network components and remove those unwanted data from the measured results.
