Normalized frequency (signal processing)

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Normalized frequency (signal processing)

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In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency ( $f$ ) and a constant frequency associated with a system (such as a sampling rate, $f_{s}$ ). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.

Examples of normalization

[edit]

A typical choice of characteristic frequency is the sampling rate ( $f_{s}$ ) that is used to create the digital signal from a continuous one. The normalized quantity, $f'={\tfrac {f}{f_{s}}},$ has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when $f$ is expressed in Hz (cycles per second), $f_{s}$ is expressed in samples per second.^[1]

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency $(f_{s}/2)$ as the frequency reference, which changes the numeric range that represents frequencies of interest from $\left[0,{\tfrac {1}{2}}\right]$ cycle/sample to $[0,1]$ half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of ${\tfrac {f_{s}}{N}},$ for some arbitrary integer $N$ (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by ${\tfrac {f_{s}}{N}}.$ ^[2]^{: p.56 eq.(16)}^[3] The normalized Nyquist frequency is ${\tfrac {N}{2}}$ with the unit ⁠1/N⁠^th cycle/sample.

Angular frequency, denoted by $\omega$ and with the unit radians per second, can be similarly normalized. When $\omega$ is normalized with reference to the sampling rate as $\omega '={\tfrac {\omega }{f_{s}}},$ the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequency for $f=1$ kHz, $f_{s}=44100$ samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:


Quantity	Numeric range	Calculation	Reverse
$f'={\tfrac {f}{f_{s}}}$	$[$ 0, ⁠1/2⁠ $]$ cycle/sample	1000 / 44100 = 0.02268	$f=f'\cdot f_{s}$
$f'={\tfrac {f}{f_{s}/2}}$	[0, 1] half-cycle/sample	1000 / 22050 = 0.04535	$f=f'\cdot {\tfrac {f_{s}}{2}}$
$f'={\tfrac {f}{f_{s}/N}}$	$[$ 0, ⁠N/2⁠ $]$ bins	1000 × $N$ / 44100 = 0.02268 $N$	$f=f'\cdot {\tfrac {f_{s}}{N}}$
$\omega '={\tfrac {\omega }{f_{s}}}$	[0, π] radians/sample	1000 × 2π / 44100 = 0.14250	$\omega =\omega '\cdot f_{s}$

References

[edit]

^ Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.
^ Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837. S2CID 426548.
^ Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies.

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Normalized frequency (signal processing)

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In digital signal processing (DSP), normalized frequency is a dimensionless representation of signal frequency scaled relative to the sampling frequency, enabling analysis that is independent of specific sampling rates. It is typically expressed either in cycles per sample, where the value ranges from 0 to 1 (with 0.5 corresponding to the Nyquist frequency, half the sampling rate), or in radians per sample, ranging from 0 to

2\pi

(with

\pi

at the Nyquist frequency).^[1]^[2] This normalization arises from the discrete-time nature of digital signals, where the sampling process maps continuous frequencies to a periodic spectrum, and the Nyquist frequency marks the upper limit for unique representation without aliasing. The normalized angular frequency

\omega

is defined as

\omega = 2\pi f / f_s

, where

f

is the physical frequency in Hz and

f_s

is the sampling frequency in Hz, ensuring that frequency responses of systems like linear time-invariant (LTI) filters are evaluated on the unit circle in the z-plane via

H(e^{j\omega})

.^[3]^[2] Normalized frequency is essential for designing and analyzing DSP components, such as finite impulse response (FIR) and infinite impulse response (IIR) filters, where specifications like cutoff frequencies are provided in normalized units to maintain generality across applications. For instance, in filter design, a low-pass filter might have a normalized cutoff at 0.2 cycles per sample, corresponding to 20% of the Nyquist frequency regardless of the actual

f_s

. It also standardizes the interpretation of spectra in tools like the discrete Fourier transform (DFT), where frequency bins are inherently normalized as

k/N

(for

k = 0

N-1

), facilitating efficient computation and visualization.^[4]^[5] The use of normalized frequency promotes portability in DSP algorithms, allowing the same digital filter coefficients to be applied at different sampling rates by simple rescaling, and it underpins key theorems like the Nyquist-Shannon sampling theorem in discrete contexts. In practice, software libraries such as MATLAB's Signal Processing Toolbox default to normalized frequency units for functions like freqz, which compute and plot responses over 0 to

\pi

radians per sample.^[1]

Fundamentals

Definition

In digital signal processing, normalized frequency refers to a dimensionless measure of frequency obtained by scaling the absolute frequency of a signal by the sampling frequency, thereby making it independent of the particular sampling rate employed in the digitization process.^[6] This scaling transforms the frequency into a relative quantity, typically ranging from 0 to 0.5 in cycles per sample (or 0 to π in radians per sample) for positive frequencies up to the Nyquist frequency; the full periodic range is 0 to 1 (or 0 to 2π), where 0.5 cycles per sample (or π radians per sample) corresponds to the Nyquist frequency, half the sampling rate. Normalization is employed to standardize signal analysis across diverse sampling rates, facilitating comparisons and designs that are not tied to specific hardware or acquisition parameters. It simplifies the specification of digital filters and other processing algorithms by allowing frequency responses to be described in proportional terms, such as fractions of the sampling rate, rather than absolute values that vary with sampling conditions.^[7] In contrast to absolute frequency, which is quantified in hertz (Hz) and reflects the physical oscillation rate in continuous time, normalized frequency lacks units and emphasizes the signal's behavior relative to the discrete sampling framework.^[8] This unitless nature underscores its role in discrete-time domains, where the sampling process inherently bounds the representable frequencies. The concept of normalized frequency arose alongside the broader emergence of digital signal processing during the 1960s and 1970s, a period when advancing digital computing capabilities first enabled practical discrete-time analysis of signals previously handled in the analog domain.^[9] Seminal works, such as those by Oppenheim and Schafer in their 1975 textbook Digital Signal Processing, formalized its use in theoretical and applied contexts.

Mathematical Representation

In digital signal processing, the normalized frequency

f_n

is defined as the ratio of the absolute frequency

f

(in hertz) to the sampling frequency

f_s

(also in hertz), yielding a dimensionless measure in cycles per sample:

f_n = \frac{f}{f_s}

This formulation arises from the sampling process, where a continuous-time exponential signal

x(t) = e^{j 2\pi f t}

is discretized to

x = x(n T_s) = e^{j 2\pi f n / f_s} = e^{j 2\pi f_n n}

, with

T_s = 1/f_s

as the sampling period, directly scaling the frequency domain by the inverse of the sampling rate.^[10] An equivalent representation uses the normalized angular frequency

\omega_n

in radians per sample, obtained by multiplying the normalized frequency by

2\pi

\omega_n = 2\pi f_n = \frac{2\pi f}{f_s}

This angular form is standard in the discrete-time Fourier transform (DTFT), where the transform

X(e^{j \omega_n}) = \sum_{n=-\infty}^{\infty} x e^{-j \omega_n n}

is periodic with period

2\pi

, reflecting the inherent repetition in discrete-time spectra.^[11] For unambiguous representation of bandlimited signals, the normalized frequency

f_n

typically ranges from 0 to 0.5, corresponding to absolute frequencies from 0 Hz to the Nyquist frequency

f_s/2

; the full spectrum, accounting for negative frequencies, extends from -0.5 to 0.5. In angular terms,

\omega_n

ranges from 0 to

\pi

(or

-\pi

\pi

) over this interval. These boundaries stem from the sampling theorem, ensuring no spectral overlap in the principal period.^[12] The derivation from continuous- to discrete-time signals highlights aliasing implications at these boundaries: the DTFT's periodicity implies that

e^{j \omega_n n} = e^{j (\omega_n + 2\pi k) n}

for any integer

k

, so absolute frequencies

f + k f_s

(for integer

k

) map to the same

f_n \mod 1

. If the continuous signal's bandwidth exceeds

f_s/2

, higher frequencies alias into the principal range

|f_n| \leq 0.5

, distorting the discrete spectrum; at exactly

f_n = 0.5

(or

\pi

radians), positive and negative Nyquist components coincide, potentially causing folding artifacts.^[7]

Normalization in Sampling

Sampling Frequency and Nyquist Limit

In digital signal processing, the sampling frequency

f_s

represents the rate at which a continuous-time analog signal is discretized into a sequence of samples, typically measured in hertz (Hz) as the number of samples acquired per second.^[13] The Nyquist-Shannon sampling theorem establishes the fundamental requirement for accurate signal reconstruction, stating that a bandlimited continuous-time signal with maximum frequency component

f_{\max}

must be sampled at a rate

f_s \geq 2 f_{\max}

to avoid aliasing and enable perfect recovery of the original signal using an ideal low-pass filter.^[14] This theorem, originally formulated by Harry Nyquist in 1928 and rigorously proved by Claude Shannon in 1949, ensures that the sampling process captures all necessary information without loss.^[15]^[16] The Nyquist frequency, defined as

f_N = \frac{f_s}{2}

, denotes the highest frequency that can be accurately represented in the sampled signal without aliasing, serving as the critical upper limit for the signal's bandwidth.^[14] Sampling at exactly

f_N

preserves all information from the bandlimited signal, while rates above this provide no additional benefit for reconstruction.^[17] Undersampling, where

f_s < 2 f_{\max}

, leads to aliasing, in which higher-frequency components fold back into the lower-frequency range, causing irreversible distortion known as frequency folding.^[18] In the context of normalized frequency (expressed as a fraction of

f_s

), this folding manifests as frequencies exceeding 0.5 wrapping around to appear as aliases below 0.5, preventing unambiguous recovery of the original spectrum.^[18]

Normalization Process

The normalization process for frequencies in sampled signals involves converting absolute (physical) frequencies to a dimensionless form that is independent of the specific sampling rate, facilitating analysis and design in digital signal processing. The procedure starts by identifying the absolute frequency

f

of the signal component of interest, typically measured in hertz (Hz). Next, the sampling frequency

f_s

, defined as the number of samples taken per second, is established based on the system's requirements. The normalized linear frequency

f_n

is then calculated using the formula

f_n = \frac{f}{f_s},

which expresses the frequency in cycles per sample and typically ranges from 0 to 1 for the unique representation within one sampling period.^[19]^[8] In some contexts, particularly when working with angular representations, the normalization adjusts for radians per sample. Here, the absolute angular frequency

\omega = 2\pi f

(in radians per second) is first considered, and the normalized angular frequency

\omega_n

is obtained as

\omega_n = \frac{\omega}{f_s} = 2\pi \frac{f}{f_s} = 2\pi f_n,

with

\omega_n

ranging from 0 to

2\pi

radians per sample. This convention is common in derivations involving the discrete-time Fourier transform (DTFT), where the transform is periodic with period

2\pi

.^[8] Due to the inherent periodicity introduced by sampling, normalized frequencies are evaluated modulo 1 in the linear scale (or modulo

2\pi

in the angular scale), reflecting the replication of the spectrum every

f_s

Hz. For instance, a normalized frequency of 1.1 cycles per sample is equivalent to 0.1 cycles per sample, as higher values alias back into the principal range. This periodicity ensures that frequencies outside [0, 1) are folded into the baseband, emphasizing the need to consider the full periodic extensions during analysis.^[19] Practical considerations in applying this process include selecting an appropriate

f_s

to achieve the desired resolution in the normalized domain. A higher

f_s

allows for finer granularity in distinguishing closely spaced frequencies when mapped to

f_n

, while still maintaining the normalized scale's invariance. To prevent aliasing, the absolute frequency must satisfy

f < f_s / 2

(the Nyquist frequency

f_N

), ensuring

f_n < 0.5

; exceeding this limit causes higher frequencies to fold into lower ones via the modulo operation.^[19] The choice between linear and angular normalization conventions depends on the application: linear normalization (dividing by

f_s

) is prevalent in spectrum plotting and filter specifications for its intuitive cycles-per-sample interpretation, whereas angular normalization (dividing by

f_s / 2\pi

, or equivalently scaling by

2\pi

) aligns with phase-based computations and trigonometric identities in algorithms like the fast Fourier transform.^[8]

Applications in Digital Signal Processing

Filter Design

In digital filter design, normalized frequency plays a crucial role in specifying filter characteristics, including cutoff frequencies, passband edges, and stopband edges, which are typically expressed as fractions of the Nyquist frequency. For example, a low-pass filter might be specified with a cutoff frequency of

f_n = 0.3

, indicating that the passband extends to 30% of the Nyquist limit, allowing designers to define requirements independently of the specific sampling rate. This approach facilitates the use of standardized design tools and algorithms that operate on a unit-normalized frequency scale from 0 to 1.^[20]^[1] A key advantage of using normalized frequency is the invariance property of the resulting filter coefficients, which remain identical regardless of the actual sampling frequency

f_s

, as long as all frequency specifications are provided in normalized units. This property simplifies the design process, enabling a single set of coefficients to be applied across different sampling rates by simply rescaling the input and output signals appropriately, without redesigning the filter. It stems from the inherent scaling in the discrete-time Fourier transform, where frequency responses are inherently relative to the sampling rate.^[21]^[22] For infinite impulse response (IIR) filter design, techniques such as the bilinear transform leverage normalized frequency through prewarping to map analog prototypes from the s-plane to the digital z-plane while preserving critical frequency points. Prewarping adjusts the analog frequencies to account for the nonlinear compression of the frequency axis in the bilinear mapping, using the relation

\omega_a = 2 \tan(\pi f_n)

, where

\omega_a

is the prewarped analog angular frequency and

f_n

is the normalized digital frequency. This ensures that the digital filter accurately reproduces the analog response at specified normalized frequencies, such as passband or stopband edges.^[23]^[24] Both finite impulse response (FIR) and IIR filters commonly employ normalized frequency grids during synthesis. In FIR design, methods like the windowing technique or frequency sampling method compute the impulse response coefficients by evaluating the desired frequency response on a normalized grid from 0 to

\pi

radians per sample, ensuring linear phase and precise control over the magnitude response. For IIR filters, normalized frequencies guide pole-zero placement in the z-plane to achieve stability and meet specifications, often building on analog designs transformed via bilinear methods. These approaches, as detailed in standard digital signal processing references, enable efficient implementation in tools like MATLAB's Signal Processing Toolbox.^[20]^[25]

Spectral Analysis

In spectral analysis, normalized frequency plays a central role in the discrete Fourier transform (DFT), where the frequency bins are indexed by

k

(for

k = 0, 1, \dots, N-1

), corresponding to normalized frequencies

f_n = \frac{k}{N}

in cycles per sample, or equivalently

\omega_n = 2\pi \frac{k}{N}

in radians per sample, with

N

denoting the DFT length.^[26]^[27] This normalization ensures that the frequency axis spans from 0 to nearly 1 (or 0 to

2\pi

in radians), representing the full range up to the sampling frequency, independent of the actual sampling rate

f_s

.^[28] For power spectral density (PSD) estimation, the normalized frequency axis facilitates plotting the magnitude and phase of the DFT output, allowing direct comparison across signals sampled at different rates without rescaling.^[28] The PSD is typically computed as the squared magnitude of the DFT coefficients, scaled by

1/N

for unbiased estimation, and displayed on a normalized scale where the axis invariance to

f_s

highlights inherent signal characteristics like dominant frequencies.^[27] This approach is particularly useful in methods like the periodogram or Welch's averaged periodogram, where the normalized bins provide a consistent framework for spectral visualization.^[29] Windowing in DFT-based spectral analysis influences frequency resolution, defined in normalized terms as

\Delta f_n = \frac{1}{N}

cycles per sample, which determines the spacing between resolvable frequency components.^[27] Applying windows, such as the Hamming window, broadens the main lobe (e.g., to 4 bins wide) while suppressing sidelobes, thereby trading finer resolution for reduced spectral leakage, with the normalized resolution remaining tied to

N

rather than physical units.^[26] Zero-padding can interpolate the spectrum to refine apparent resolution without altering the underlying

\Delta f_n

.^[29] Spectral leakage and aliasing effects are effectively visualized on the normalized frequency scale, typically ranging from -0.5 to 0.5 cycles per sample to capture the Nyquist interval symmetrically.^[28] Leakage manifests as energy spillover from a sinusoid into adjacent bins due to finite

N

, mitigated by windowing that lowers sidelobe levels (e.g., Hamming reduces them to about -40 dB), while aliasing folds frequencies beyond 0.5 back into the principal range, observable as mirrored artifacts in the plot.^[27] This normalized representation aids in diagnosing these phenomena without dependence on specific hardware sampling rates.^[26]

Examples

Audio Signal Normalization

In audio signal processing, normalized frequency provides a sampling-rate-independent way to represent tones within the audible spectrum. For instance, in a standard compact disc audio signal sampled at 44.1 kHz, a pure 1 kHz tone corresponds to a normalized frequency of

f_n = \frac{1000}{44100} \approx 0.023

cycles per sample.^[30] The typical human hearing range of 20 Hz to 20 kHz normalizes to approximately 0 to 0.45 in this context, approaching but not exceeding the Nyquist limit of 0.5, ensuring faithful representation without aliasing. This normalization proves particularly useful in audio equalizers, where cutoff frequencies for bass or treble adjustments are often specified in normalized terms to maintain consistent behavior across different sampling rates, enabling resampling invariance. For example, a low-shelf filter cutoff at a normalized frequency of 0.05 (corresponding to roughly 2.2 kHz at 44.1 kHz sampling) will proportionally adjust when the signal is resampled to 48 kHz, preserving the perceptual balance without recalibration.^[31] Such designs are common in parametric equalizers implemented in digital audio workstations, relying on the fraction-of-sampling-rate metric to ensure portability.^[32] A practical case study arises in MP3 compression, where perceptual coding at low bitrates can introduce artifacts manifesting as aliasing-like distortions. The MP3 algorithm employs a polyphase filterbank to divide the signal into subbands, introducing inter-subband aliasing that is canceled during reconstruction, but imperfect quantization at low bitrates can lead to residual artifacts, particularly in high-frequency regions above 16 kHz, manifesting as metallic or ringing sounds in the audible band.^[33] These effects become evident when analyzing the normalized spectrum of compressed audio, highlighting the importance of adequate bitrate to mitigate such artifacts within the 0 to 0.5 range. Tools like MATLAB or Octave facilitate computation of normalized spectra for audio files, aiding in visualization and analysis. The following code snippet loads a WAV file, computes its discrete Fourier transform (DFT), and plots the magnitude spectrum in normalized frequency units:

matlab

% Load audio file [y, Fs] = audioread('example_audio.wav'); % Fs is sampling frequency, e.g., 44100 Hz % Compute DFT N = [length](/page/Length)(y); Y = fft(y); f_normalized = (0:N/2-1) / (N/2); % Normalized frequency: 0 to 1 (Nyquist = 1) % Plot single-sided magnitude spectrum plot(f_normalized, 2*abs(Y(1:N/2))/N); xlabel('Normalized Frequency (×π rad/sample)'); ylabel('Magnitude'); title('Normalized Spectrum of Audio Signal'); grid on;

This approach normalizes the frequency axis relative to the Nyquist frequency, with the plot spanning 0 to 1 for clarity in audio diagnostics.^[34]

Image Processing Normalization

In image processing, normalized frequency extends the one-dimensional concept to two-dimensional spatial domains, where it quantifies the rate of variation in pixel intensities along horizontal and vertical directions relative to the sampling rates. For a digital image, the normalized spatial frequencies are defined as

f_{n_x} = \frac{f_x}{f_{s_x}}

and

f_{n_y} = \frac{f_y}{f_{s_y}}

, with

f_x

and

f_y

representing the actual spatial frequencies in cycles per unit length (e.g., cycles per inch), and

f_{s_x}

and

f_{s_y}

denoting the sampling frequencies in samples per unit length (e.g., pixels per inch).^[35] These normalized values range from 0 to 0.5, corresponding to direct current (DC) to the Nyquist frequency, facilitating device-independent analysis of image sharpness and detail preservation. Consider a 512×512 pixel image scanned at 300 dots per inch (dpi), yielding sampling rates

f_{s_x} = f_{s_y} = 300

pixels per inch. A horizontal line pattern repeating at 50 cycles per inch has an actual frequency

f_x = 50

cycles per inch, resulting in a normalized frequency

f_{n_x} = \frac{50}{300} \approx 0.167

cycles per pixel; the vertical component

f_{n_y} = 0

since the pattern lacks vertical variation.^[35] This normalization highlights that the pattern occupies about one-third of the Nyquist limit (0.5 cycles per pixel), aiding in assessing resolution limits without specifying absolute units.^[35] Normalized frequencies are essential in designing image filters within the frequency domain, particularly using the two-dimensional fast Fourier transform (2D FFT) to apply low-pass or high-pass operations. For instance, a low-pass filter with a normalized cutoff at 0.2 cycles per pixel attenuates higher spatial frequencies to blur the image, reducing noise while preserving low-frequency structures like edges; conversely, a high-pass filter with a cutoff at 0.3 sharpens details by emphasizing mid-to-high frequencies.^[35] These cutoffs are specified in normalized terms to ensure filter responses scale appropriately across images of varying resolutions, as implemented in standard libraries like MATLAB's Image Processing Toolbox.^[36] In cases of anisotropic sampling, such as scanned documents where horizontal and vertical resolutions differ (e.g., 600 dpi horizontally and 300 dpi vertically due to mechanical constraints), separate normalizations

f_{n_x} = \frac{f_x}{600}

and

f_{n_y} = \frac{f_y}{300}

account for the unequal sampling rates

f_{s_x} \neq f_{s_y}

.^[37] This approach prevents distortion in frequency-domain processing, ensuring that filter designs or modulation transfer function (MTF) analyses reflect the true spatial response; for example, a vertical pattern at 100 cycles per inch yields

f_{n_y} = \frac{100}{300} \approx 0.333

, higher than its horizontal counterpart at the same physical frequency.^[37]

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Normalized frequency (signal processing)

Fundamentals

Definition

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Filter Design

Spectral Analysis

Examples

Audio Signal Normalization

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Normalized frequency (signal processing)

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Normalization in Sampling

Sampling Frequency and Nyquist Limit

Normalization Process

Applications in Digital Signal Processing

Filter Design

Spectral Analysis

Examples

Audio Signal Normalization

Image Processing Normalization

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