Filtering and Signal Analysis

The Central Idea

For a linear time-invariant system with impulse response \(h(t)\), the output is obtained by convolution:

Taking the Fourier transform converts convolution into multiplication:

This pair of equations is the organizing principle of filtering. In the time domain, a filter combines shifted copies of a signal. In the frequency domain, the same filter multiplies each spectral component by a frequency-dependent gain and phase shift.

What Is a Signal?

A signal is a quantity that varies with an independent variable such as time, position, sample number, or frequency. Examples include voltage over time, sound pressure, temperature measurements, pixel intensity, vibration data, financial measurements, and sensor outputs.

A continuous-time signal may be written

while a sampled signal is commonly written

The mathematical methods differ slightly between continuous and discrete settings, but the central questions remain the same: what components are present, which are useful, and how should they be modified?

What Does a Filter Do?

A filter changes some components of a signal more than others. Depending on its design, a filter may:

• preserve slowly varying behavior while suppressing rapid fluctuations;
• remove low-frequency drift while retaining fast changes;
• isolate a selected band of frequencies;
• reject a narrow interfering frequency;
• change phase or timing without greatly changing amplitude;
• reduce measurement noise;
• emphasize transitions, edges, or periodic structure;
• prepare data for later estimation, detection, or modeling.

Filtering therefore means more than simply making a signal smoother. It is a deliberate redistribution of spectral content.

Impulse Response

The impulse response \(h(t)\) describes how a linear time-invariant system responds to an ideal impulse. Once \(h(t)\) is known, the response to a general input is

This is convolution. The function \(h\) acts as the time-domain description of the filter.

For a discrete signal,

In finite computations, only the nonzero or retained samples of \(h[k]\) need to be included.

Frequency Response

The Fourier transform of the impulse response is called the frequency response:

If the input has transform \(X(\omega)\), then

Thus, \(H(\omega)\) tells us exactly how the filter acts on each frequency. A frequency for which \(|H(\omega)|\) is near \(1\) is largely preserved. A frequency for which \(|H(\omega)|\) is near \(0\) is strongly suppressed.

Magnitude and Phase Response

A complex frequency response can be written in polar form as

The magnitude response \(|H(\omega)|\) controls amplitude. The phase response \(\phi(\omega)\) controls phase shift.

If a sinusoidal input is

then a stable LTI filter produces a steady-state sinusoid at the same frequency:

The filter changes the amplitude and phase, but it does not create a new frequency from a single sinusoidal input when the system is linear and time-invariant.

Passbands and Stopbands

A passband is a frequency region intended to pass through a filter with relatively little attenuation. A stopband is a region intended to be strongly suppressed.

Real filters usually contain a transition region between passband and stopband. Idealized diagrams may show a perfectly sharp boundary, but exact sharp transitions generally require nonlocal or infinite-duration impulse responses.

Low-Pass Filters

A low-pass filter preserves lower frequencies and attenuates higher frequencies. It is commonly used for smoothing, noise reduction, trend extraction, and anti-aliasing.

An ideal low-pass response with cutoff \(\omega_c\) can be written

Frequencies below the cutoff are passed, while frequencies above it are removed. Practical filters approximate this behavior with a smoother transition.

High-Pass Filters

A high-pass filter does the opposite: it suppresses slowly varying components and preserves more rapidly varying components.

High-pass filtering is useful for removing baseline drift, emphasizing rapid changes, detecting edges, and isolating high-frequency vibration or transient content.

An idealized high-pass response is

Band-Pass Filters

A band-pass filter preserves frequencies inside a selected interval and suppresses frequencies below and above it.

An ideal symmetric band-pass filter may be described by

for frequencies satisfying

and \(H(\omega)=0\) elsewhere.

Band-pass filtering is useful when a signal of interest occupies a known frequency band, such as a communication channel, vibration mode, or acoustic range.

Band-Stop and Notch Filters

A band-stop filter rejects a selected interval while preserving frequencies outside that interval. A notch filter is a narrow band-stop filter designed to suppress a small frequency neighborhood.

Notch filters are useful when interference is concentrated near a specific frequency. The design challenge is to reject the unwanted component without unnecessarily disturbing nearby useful frequencies.

Cutoff Frequency and Bandwidth

The cutoff frequency marks a transition between a region intended to pass and a region intended to attenuate. Different engineering conventions may define the cutoff at different gain levels, so a cutoff should be interpreted together with the filter specification being used.

For a band extending from \(f_1\) to \(f_2\), the bandwidth is

Narrow bandwidth isolates a smaller range of frequencies. Wide bandwidth preserves a broader range and usually retains faster time-domain changes.

A First-Order Low-Pass Model

A common first-order low-pass frequency response is

where \(\tau>0\) is a time constant.

Its magnitude is

and its phase is

When \(|\omega|\tau\ll1\), the gain is near \(1\). When \(|\omega|\tau\gg1\), the magnitude becomes small. This creates a gradual rather than perfectly sharp transition from passband to stopband.

Time Constant and Frequency Scale

The parameter \(\tau\) controls how quickly the first-order system responds in time. It also determines the characteristic angular frequency

In ordinary frequency,

A larger time constant means a slower response and a lower characteristic frequency. This is an important example of the reciprocal relationship between time scale and frequency scale.

Filtering by Convolution

Frequency-domain multiplication and time-domain convolution describe the same operation:

This duality gives two implementation strategies. If the impulse response is short, direct convolution may be efficient. If the data and filter are long, transform methods can reduce the computational cost by converting convolution into pointwise multiplication.

A Simple Discrete Smoothing Filter

Consider the three-point moving average

Its impulse response is

with all other samples equal to zero.

The output is a local average, so isolated rapid changes are reduced. This makes the filter low-pass in character.

The Moving-Average Frequency Response

For an \(M\)-point moving average,

Its discrete-time frequency response is

This may be rewritten as

The magnitude is largest near \(\Omega=0\), confirming the low-pass behavior. Zeros occur at selected nonzero frequencies, so some oscillatory components can be canceled exactly.

Smoothing Comes with a Tradeoff

Increasing the width of a moving average usually produces stronger smoothing, but it also spreads abrupt changes over a wider interval and can reduce temporal resolution.

This is a recurring principle in signal processing:

A useful filter therefore balances noise reduction against distortion of the desired signal.

Difference Filters and Rapid Change

The first difference

has impulse response

Its frequency response is

Since \(H(0)=0\), a constant component is removed. The filter therefore emphasizes changes rather than steady levels and behaves like a simple high-pass or discrete differentiation operator.

Filtering and Differentiation

In the continuous Fourier transform,

Differentiation multiplies the spectrum by \(|\omega|\), so higher frequencies are amplified more strongly than lower frequencies.

This is useful for detecting rapid changes, but it also explains why numerical differentiation is highly sensitive to high-frequency noise. In practice, a smoothing operation is often combined with differentiation.

Signal and Noise

A measured quantity is often modeled as

where \(s(t)\) is the desired signal and \(n(t)\) is unwanted noise or interference.

Filtering is most effective when the useful signal and the unwanted component have different spectral characteristics. If the signal occupies low frequencies while the noise is concentrated at high frequencies, a low-pass filter can improve the measurement substantially. If their spectra overlap strongly, no simple frequency filter can separate them perfectly.

Signal-to-Noise Ratio

One way to quantify signal quality is the signal-to-noise ratio

where \(P_s\) and \(P_n\) represent suitable signal and noise powers.

In decibels,

A filter can improve SNR if it attenuates noise-dominated spectral regions more than signal-dominated regions.

Analyzing a Signal with the Fourier Transform

A Fourier transform reveals which frequencies contribute to a continuous signal. For sampled data, the DFT provides a finite spectral representation:

The magnitude \(|X[k]|\) indicates the strength of the corresponding frequency-bin component, while the argument of \(X[k]\) gives phase information.

Large peaks in a spectrum often indicate dominant periodic components, resonances, harmonics, or interference frequencies.

Amplitude Spectra and Power Spectra

The amplitude spectrum emphasizes component amplitudes, usually through quantities derived from \(|X[k]|\). A simple finite-record power measure is proportional to

With an appropriate normalization, this produces a periodogram-type estimate of how energy or power is distributed among frequency bins.

The precise scaling depends on the DFT convention, sample interval, record length, and whether a one-sided or two-sided spectrum is being reported. Consistent normalization is therefore essential when comparing spectral magnitudes.

Parseval's Principle in Signal Analysis

Parseval-type identities express the fact that total energy can be computed in either the original domain or the transform domain.

With the DFT convention

Parseval's identity is

Filtering redistributes or attenuates these spectral contributions according to \(|H[k]|\).

Windowing a Finite Record

Real measurements are observed for a finite duration. Multiplying a longer signal by a finite observation window modifies its spectrum. If the measured record does not contain an integer number of periods of a sinusoid, spectral energy can spread into neighboring DFT bins.

This spreading is called spectral leakage.

A window function \(w[n]\) forms the analyzed sequence

Different windows trade narrow spectral peaks against reduced sidelobes. Windowing does not create additional information; it changes how the finite record distributes spectral leakage.

Frequency Resolution

If \(N\) samples are collected at sampling rate \(f_s\), the DFT bin spacing is

Since the observation duration is

the same relation can be written

Longer observation time provides finer frequency-bin spacing. This is another form of the time-frequency tradeoff: resolving nearby frequencies generally requires a sufficiently long record.

Sampling and the Nyquist Limit

A sampling rate \(f_s\) can represent frequencies only up to the Nyquist frequency

Components above this limit can appear at incorrect lower frequencies after sampling. This effect is called aliasing.

Once aliasing has occurred, the sampled data alone generally cannot distinguish the aliased component from a genuine lower-frequency component.

Anti-Aliasing Filters

Before analog data are sampled, a low-pass filter is often used to attenuate frequency components above the intended Nyquist range. This is an anti-aliasing filter.

The workflow is conceptually

The filter transition band must be accounted for in the choice of sampling rate; practical filters do not change from full transmission to zero transmission at one exact frequency.

Filtering Sampled Data

Once data are sampled, filtering may be performed directly in the time domain or in the DFT domain.

A frequency-domain workflow is

1. compute the DFT of the data;
2. construct or evaluate a spectral response \(H[k]\);
3. form \(Y[k]=H[k]X[k]\);
4. compute the inverse DFT.

For linear convolution with finite sequences, suitable zero padding must be used to prevent unintended circular wraparound.

Finite Impulse Response Filters

A finite impulse response, or FIR, filter has only finitely many nonzero coefficients. A common form is

FIR filters are conceptually straightforward because they are finite convolutions. Their frequency response is

By selecting the coefficients \(b_k\), we shape the magnitude and phase response.

Recursive Filters

Some discrete filters also use previous output values. For example,

This is a simple recursive smoothing filter. Unlike a finite moving average, its impulse response continues indefinitely, though it decays when the parameters are stable.

Recursive filters can achieve strong frequency selectivity with relatively few stored coefficients, but stability and phase behavior require careful analysis.

Filter Stability

A useful input-output notion of stability is bounded-input, bounded-output stability: every bounded input should produce a bounded output.

For an LTI system, absolute integrability of the continuous impulse response,

is a standard sufficient-and-necessary criterion for BIBO stability under the usual LTI assumptions. In discrete time, the analogous condition is

Stability matters because a filter that amplifies certain internal modes without bound can overwhelm the intended signal processing operation.

Phase Distortion and Delay

Two filters can have similar magnitude responses but produce different waveforms because their phase responses differ.

A pure time delay \(t_0\) has frequency response

Its magnitude is \(1\) at every frequency, but its phase is

Linear phase therefore corresponds to uniform delay. Nonlinear phase can shift different frequency components by different effective amounts and distort the shape of a broadband waveform.

Filtering in Spectral Coordinates

The same idea extends beyond ordinary Fourier frequency. Suppose a signal or state has an expansion

in orthogonal spectral modes \(\phi_k\). A spectral filter acts by changing the mode coefficients:

The multipliers \(g_k\) play the same role as the frequency response \(H(\omega)\). This connects ordinary signal filtering directly to eigenfunction expansions and spectral decomposition.

Mode Truncation as a Filter

A particularly simple spectral filter keeps only selected modes. If

then a truncated approximation is

If higher-index modes correspond to increasingly rapid oscillation, truncation acts like a low-pass operation in the operator's natural spectral coordinates. This idea appears in Fourier approximation, PDE solvers, model reduction, and denoising.

Inverse Filtering and Its Limits

If

then one might try to recover the input through

This is possible only where \(H(\omega)\neq0\), and it can be highly unstable when \(|H(\omega)|\) is very small. Noise in a strongly attenuated band may be amplified enormously by division.

Deconvolution and inverse problems therefore often require regularization rather than naive spectral division.

Detecting Dominant Frequencies

Suppose a sampled signal contains several sinusoidal components. A practical analysis workflow is to compute a windowed DFT and inspect the magnitude or power spectrum for peaks.

A peak near frequency \(f_0\) suggests a strong component near that frequency, but interpretation should also consider:

• frequency resolution;
• window shape;
• spectral leakage;
• noise floor;
• harmonics and sidebands;
• possible aliasing;
• whether the signal remains approximately stationary over the record.

Harmonics and Periodic Structure

A periodic but nonsinusoidal signal often produces spectral peaks at an integer sequence of frequencies related to a fundamental frequency \(f_0\):

These components are called harmonics. Their relative strengths help characterize waveform shape. Filtering selected harmonics can change that shape even when the fundamental period remains unchanged.

Transient Signals Need Time Information

A single Fourier spectrum summarizes frequency content over the analyzed record, but it does not indicate exactly when each component occurred.

If a signal changes character over time, a local or windowed analysis can be more informative. One strategy is to analyze overlapping time windows, producing a time-frequency representation.

Short windows provide better temporal localization but poorer frequency resolution; long windows do the reverse. This is the practical manifestation of the time-frequency localization tradeoff developed earlier.

Example: Removing High-Frequency Noise

Suppose

The first term varies slowly and the second much more rapidly. If the second term is unwanted interference, a low-pass filter with

and

produces approximately

The filter succeeds because the desired and unwanted components are spectrally separated.

Example: First-Order Filter Gain

Let

and consider an input at \(\omega=3\). The magnitude response is

Therefore, an input sinusoid of amplitude \(5\) has steady-state output amplitude

The phase shift is

Example: Three-Point Moving Average

Let the data be

Using the causal three-point moving average and assuming earlier samples are zero,

we obtain

and

The output rises more gradually because each value blends information from several neighboring samples.

Example: First Difference

For the sequence

define

with \(x[-1]=0\). Then

Instead of representing the level of the original signal, the output represents its sample-to-sample change.

Example: Frequency Resolution

Suppose \(N=2000\) samples are collected at

The record length is

Therefore, the DFT bin spacing is

The same result follows from \(\Delta f=1/T\).

Common Error: Treating Every Filter as an Ideal Cutoff

Real filters usually have transition bands. A statement such as "everything below \(100\) Hz passes and everything above \(100\) Hz disappears" describes an idealized model, not a typical realizable response.

Always inspect the actual magnitude response and the attenuation specification.

Common Error: Looking Only at Magnitude

Magnitude response tells how strongly each frequency is amplified or attenuated, but phase response affects timing and waveform shape.

When waveform preservation matters, magnitude alone is not sufficient.

Common Error: Filtering After Aliasing

Digital filtering cannot generally recover information that was irreversibly folded into the wrong frequency band during sampling.

High-frequency components should be controlled before sampling when aliasing is a concern. That is the purpose of anti-aliasing filters.

Common Error: Ignoring Spectral Leakage

A broad group of neighboring DFT bins does not automatically mean that the original signal contains many distinct frequencies. A single sinusoid observed over a finite noninteger number of periods can spread energy across bins.

Inspect the observation duration and windowing strategy before interpreting every spectral bin as a separate physical component.

Common Error: Confusing Bin Spacing with Guaranteed Resolution

The quantity

gives the spacing between DFT sample frequencies. It does not by itself guarantee that two nearby tones can be cleanly distinguished. Window shape, signal strength, noise, observation time, and estimator design also matter.

Common Error: Using Circular Convolution Accidentally

Multiplying two length-\(N\) DFTs and applying an inverse DFT produces circular convolution unless the sequences have been padded appropriately.

For finite sequences of lengths \(L\) and \(M\), at least

output positions are needed for ordinary linear convolution without wraparound.

A Reliable Signal-Analysis Workflow

1. identify the quantity being measured and its units;
2. determine whether the signal is continuous or sampled;
3. record the sampling rate and observation duration;
4. inspect the signal in the time domain;
5. remove known offsets or trends only when justified;
6. choose an appropriate window when using a finite DFT record;
7. compute the spectrum with a consistent normalization;
8. identify useful bands, interference, harmonics, and noise regions;
9. choose the filter type from the signal objective;
10. inspect both magnitude and phase response when relevant;
11. apply the filter with proper handling of boundaries and convolution length;
12. compare the filtered and original signals in both domains;
13. verify that desired features were preserved;
14. document cutoff frequencies, bandwidths, delays, and other design choices.

Choosing the Right Domain

Use the time domain when the timing, duration, onset, decay, or transient shape is the main concern.

Use the frequency domain when periodic components, bandwidth, resonance, harmonic structure, filtering, or spectral energy distribution is the main concern.

In many real problems, the strongest analysis moves back and forth between the two. A filter specification may be designed spectrally, implemented by convolution, and validated by inspecting the resulting waveform in time.

Practice 1: Filter Type

A measurement contains a slowly varying desired trend and strong rapid fluctuations. Which broad filter type is the natural first choice?

Practice 2: First-Order Magnitude

For

compute \(|H(1)|\).

Practice 3: Characteristic Frequency

A first-order low-pass system has time constant

Find its characteristic angular frequency \(\omega_c\).

Practice 4: Moving Average

Using

compute \(y[2]\) if

Practice 5: First Difference

For

compute \(y[2]\) when

Practice 6: Nyquist Frequency

A signal is sampled at

Find the Nyquist frequency.

Practice 7: DFT Bin Spacing

A record contains \(N=4096\) samples collected at \(f_s=2048\) Hz. Find the DFT bin spacing.

Practice 8: SNR

If the signal power is \(100\) times the noise power, find the SNR in decibels.

Practice 9: Spectral Filtering

A signal is represented as

A spectral filter uses gains

Find the filtered signal.

Practice 10: Identify the Limitation

A system has a frequency response satisfying

Explain why naive inverse filtering cannot recover the input component at \(\omega_0\) from the output.

Practice Answers

Practice 1

A low-pass filter is the natural first choice because the desired behavior is slow while the unwanted variation is rapid.

\[ \boxed{\text{Low-pass filter}} \]

Practice 2

At \(\omega=1\),

Therefore,

Practice 3

Since

we obtain

Practice 4

Thus,

Practice 5

Therefore,

Practice 6

Therefore,

Practice 7

Hence,

Practice 8

Therefore,

Practice 9

Multiply each spectral coefficient by its gain:

Thus,

Practice 10

At \(\omega_0\), the output component satisfies

The filter has completely removed that spectral component. Division by \(H(\omega_0)\) is impossible because the denominator is zero, so the original component cannot be recovered by naive inverse filtering.

What to Remember

The central LTI filtering relationships are

and

• the impulse response describes a filter in the time domain;
• the frequency response describes the same filter in the frequency domain;
• magnitude response controls amplitude scaling;
• phase response controls phase shift and timing behavior;
• low-pass filters preserve lower frequencies;
• high-pass filters preserve higher frequencies;
• band-pass filters isolate a selected frequency range;
• notch filters suppress a narrow unwanted range;
• moving averages are simple low-pass filters;
• difference operators emphasize rapid change;
• smoothing reduces high-frequency content but can blur fast features;
• DFT analysis reveals spectral components of sampled data;
• finite records introduce leakage and resolution tradeoffs;
• a sufficiently high sampling rate, together with appropriate band-limiting, is essential for avoiding aliasing;
• anti-aliasing filtering should occur before sampling when needed;
• phase can matter even when magnitude is unchanged;
• spectral filtering generalizes naturally to eigenfunction and operator modes;
• inverse filtering becomes unstable near frequencies where the filter response is small.

The main operational habit is to inspect both domains. The time-domain waveform shows when behavior occurs; the spectral representation shows which modes or frequencies are responsible for it.

Where We Go Next

Transform and spectral methods are not limited to signal processing. The same orthogonal modes that separate frequencies also arise when differential equations are constrained by conditions at spatial boundaries. Those conditions select the admissible eigenfunctions and determine which spectral modes can occur.

The next lesson, Boundary-Value Problems, develops this connection. We will examine differential equations subject to endpoint or boundary conditions, distinguish boundary-value problems from initial-value problems, study eigenvalue conditions, and show how Fourier and eigenfunction expansions turn boundary-driven differential equations into solvable spectral systems.