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Raised-cosine filter

A raised-cosine filter is a type of used in digital communication systems for , designed to confine the transmitted signal's spectrum to a specific while minimizing (ISI) between adjacent symbols. This filter achieves zero ISI by ensuring that its has nulls at multiples of the symbol period, satisfying the for distortionless transmission. The of the raised-cosine filter is characterized by a flat , a smooth cosine-shaped transition band, and a sharp cutoff, defined mathematically as T, & |f| \leq \frac{1 - \alpha}{2T} \\ \frac{T}{2} \left[ 1 + \cos\left( \frac{\pi T}{\alpha} \left( |f| - \frac{1 - \alpha}{2T} \right) \right) \right], & \frac{1 - \alpha}{2T} < |f| \leq \frac{1 + \alpha}{2T} \\ 0, & |f| > \frac{1 + \alpha}{2T} \end{cases} $$ where $ T $ is the symbol period and $ \alpha $ (0 ≤ α ≤ 1) is the [roll-off](/page/Roll-off) factor that determines the excess [bandwidth](/page/Bandwidth) beyond the minimum Nyquist [bandwidth](/page/Bandwidth) of $ 1/(2T) $.[](https://complextoreal.com/wp-content/uploads/2013/01/isi.pdf) A smaller $ \alpha $ yields a narrower [bandwidth](/page/Bandwidth) with slower decay in the [time domain](/page/Time_domain), increasing sensitivity to timing errors, while a larger $ \alpha $ widens the [bandwidth](/page/Bandwidth) but reduces [ISI](/page/ISI) and improves robustness to synchronization offsets.[](https://www.analog.com/en/resources/app-notes/an-922.html) The corresponding [impulse response](/page/Impulse_response) is $$ h(t) = \frac{\sin(\pi t / T)}{\pi t / T} \cdot \frac{\cos(\pi \alpha t / T)}{1 - (2 \alpha t / T)^2}, $$ which exhibits faster decay than a [sinc function](/page/Sinc_function), aiding practical implementation.[](https://www.analog.com/en/resources/app-notes/an-922.html) In digital modulation schemes such as quadrature phase-shift keying (QPSK), [quadrature amplitude modulation](/page/Quadrature_amplitude_modulation) (QAM), and [orthogonal frequency-division multiplexing](/page/Orthogonal_frequency-division_multiplexing) (OFDM), the raised-cosine filter is applied at the transmitter to shape pulses and at the receiver as a [matched filter](/page/Matched_filter) to maximize [signal-to-noise ratio](/page/Signal-to-noise_ratio) while rejecting [ISI](/page/ISI).[](https://www.sciencedirect.com/topics/engineering/raised-cosine-pulse) Often, to optimize performance over channels with distortions, the raised-cosine response is split into square-root raised-cosine filters at both ends of the link, ensuring the combined effect yields the full raised-cosine characteristics.[](https://www.mathworks.com/help/comm/ug/raised-cosine-filtering.html) This filter's versatility has made it a standard in telecommunications protocols, including wireless standards like [IEEE 802.11](/page/IEEE_802.11) and cellular systems.[](https://www.sciencedirect.com/topics/engineering/raised-cosine-pulse) ## Overview ### Definition and Purpose A raised-cosine filter is a specific type of Nyquist filter employed in digital communication systems for [pulse shaping](/page/Pulse_shaping), characterized by a [frequency response](/page/Frequency_response) that follows a raised-cosine curve to achieve zero [intersymbol interference](/page/Intersymbol_interference) (ISI) at the sampling instants.[](https://www.sciencedirect.com/topics/computer-science/raised-cosine-filter) This linear-phase filter ensures that the transmitted pulses are shaped such that their contributions do not overlap significantly with adjacent symbols when sampled at the [symbol rate](/page/Symbol_rate).[](https://www.sciencedirect.com/topics/computer-science/raised-cosine-filter) The primary purpose of the raised-cosine filter is to minimize [ISI](/page/ISI) in band-limited channels, where signals must be confined within a specified [bandwidth](/page/Bandwidth) to avoid [interference](/page/Interference) with other users or to comply with regulatory [spectral](/page/Spectral) masks.[](https://www.sciencedirect.com/topics/computer-science/raised-cosine-filter) By designing the pulse shape so that its tails decay appropriately and cross zero at multiples of the [symbol](/page/Symbol) period, the filter prevents the energy from one symbol from affecting the detection of neighboring symbols.[](https://ieeexplore.ieee.org/document/6773395) Additionally, it limits out-of-band emissions, thereby optimizing [spectral efficiency](/page/Spectral_efficiency) while maintaining signal integrity.[](https://www.sciencedirect.com/topics/computer-science/raised-cosine-filter) This functionality relies on the [Nyquist criterion](/page/Nyquist_criterion) for zero [ISI](/page/ISI), which stipulates that the overall channel response must exhibit vestigial symmetry in the [frequency domain](/page/Frequency_domain) around half the [symbol rate](/page/Symbol_rate) (the [Nyquist rate](/page/Nyquist_rate)), ensuring no residual [interference](/page/Interference) at [decision points](/page/Decision_Points).[](https://ieeexplore.ieee.org/document/6773395) For instance, in [binary](/page/Binary) signaling schemes, the raised-cosine filter guarantees that the received signal value at each [symbol](/page/Symbol) timing instant depends solely on the transmitted [symbol](/page/Symbol) for that interval, without contributions from preceding or following bits.[](https://www.sciencedirect.com/topics/computer-science/raised-cosine-filter) The filter's [roll-off](/page/Roll-off) factor serves as a tunable [parameter](/page/Parameter) to balance the trade-off between the sharpness of the frequency transition and the excess bandwidth beyond the minimum Nyquist requirement.[](https://www.sciencedirect.com/topics/computer-science/raised-cosine-filter) ### Historical Development The raised-cosine filter builds on the foundational [Nyquist criterion](/page/Nyquist_criterion) for zero [intersymbol interference](/page/Intersymbol_interference) ([ISI](/page/ISI)), established by [Harry Nyquist](/page/Harry_Nyquist) in his 1928 paper "Certain Topics in Telegraph Transmission Theory," which analyzed conditions for distortionless transmission in telegraph systems. The specific raised-cosine frequency response, providing a realizable approximation to the ideal sinc pulse with adjustable excess [bandwidth](/page/Bandwidth) via a [roll-off](/page/Roll-off) factor, was developed in subsequent decades to satisfy this [criterion](/page/Criterion) while addressing practical [bandwidth](/page/Bandwidth) constraints. During the 1950s and 1960s, the filter gained prominence in telephony research at Bell Laboratories, where it was applied to pulse shaping for early digital transmission systems like the T1 carrier introduced in 1962. Researchers, including Robert W. Lucky, advanced complementary techniques such as adaptive equalization to counter channel distortions, enabling effective use of raised-cosine pulses in PCM-based networks for ISI mitigation. Lucky's 1965 invention of automatic equalizers, detailed in subsequent Bell Labs publications, integrated seamlessly with raised-cosine shaping to support reliable data rates over twisted-pair lines. A 1967 Bell Labs memorandum further refined modified raised-cosine designs using hybrid integrated circuits for improved practicality in carrier systems.[](https://ethw.org/Oral-History:Robert_Lucky)[](https://telephonecollectors.info/index.php/browse/document-repository/catalogs-manuals-educational-docs-by-company/western-electric-bell-system/583-mm-67-5316-6) By the 1970s, the raised-cosine filter saw widespread adoption in digital modems and [data communication](/page/Data_communication) standards, optimizing [spectral efficiency](/page/Spectral_efficiency) in systems like early PCM [telephony](/page/Telephony) and modems operating over analog telephone networks. Its use in [pulse shaping](/page/Pulse_shaping) for T1 lines and emerging digital hierarchies facilitated the transition to integrated digital networks, with optimizations explored in IEEE publications for minimizing [bandwidth](/page/Bandwidth) while preserving [signal integrity](/page/Signal_integrity). In the [1980s](/page/1980s), the filter was embedded in [ITU-T](/page/ITU-T) recommendations for hierarchical digital interfaces and modulation schemes, such as those in the V-series for modems, ensuring compatibility across global telecommunications infrastructure.[](https://ntrs.nasa.gov/api/citations/19720021544/downloads/19720021544.pdf)[](https://www.itu.int/rec/T-REC-G.703) The 1990s marked the filter's evolution into wireless communications, driven by [digital signal processing](/page/Digital_signal_processing) advances, with root-raised-cosine variants standardized for standards like IS-95 CDMA to shape chip pulses and control [out-of-band](/page/Out-of-band) emissions. This period saw its integration into base stations and handsets for efficient [spectrum](/page/Spectrum) use in cellular networks. In the [modern era](/page/Modern_Era) as of 2025, variants of the raised-cosine filter, such as root-raised-cosine, continue to be relevant in [5G](/page/5G) New Radio (NR) systems for optional [pulse shaping](/page/Pulse_shaping) in DFT-s-OFDM uplink transmissions and enhanced OFDM waveforms, aiding spectral containment in implementations compliant with [3GPP](/page/3GPP) Release 15 and later, though the core standard primarily employs rectangular pulses with windowing.[](https://arxiv.org/pdf/1605.03731) ## Mathematical Formulation ### Impulse Response The impulse response of the raised-cosine filter, denoted as $ h(t) $, is the time-domain [representation](/page/Representation) that defines its pulse-shaping behavior in digital communications. It is given by the expression h(t) = \frac{\sin\left(\pi t / T\right)}{\pi t / T} \cdot \frac{\cos\left(\pi \alpha t / T\right)}{1 - 2 \alpha^2 (t / T)^2}, where $ T $ is the symbol period and $ \alpha $ is the [roll-off](/page/Roll-off) factor with $ 0 \leq \alpha \leq 1 $.[](https://www.analog.com/en/resources/app-notes/an-922.html) This formula arises from the inverse [Fourier transform](/page/Fourier_transform) of the corresponding frequency-domain specification, tailored to achieve controlled spectral occupancy while minimizing [intersymbol interference](/page/Intersymbol_interference) (ISI).[](https://www.analog.com/en/resources/app-notes/an-922.html) A key feature of $ h(t) $ is its value at specific points: $ h(0) = 1 $, providing a normalized peak at the center of the pulse, and $ h(\pm T) = 0 $, ensuring zero crossings at integer multiples of the symbol period.[](https://www.analog.com/en/resources/app-notes/an-922.html) These zero crossings are critical, as they prevent the tails of one pulse from interfering with the sampling of adjacent symbols, thereby eliminating [ISI](/page/ISI) when symbols are spaced by $ T $.[](https://www.analog.com/en/resources/app-notes/an-922.html) The [impulse response](/page/Impulse_response) is typically normalized such that its [peak](/page/Peak) [value](/page/Value) [at $ t](/page/AT&T) = 0 $ is unity, ensuring unity [gain](/page/Gain) in the [passband](/page/Passband) for matched filtering applications.[](https://www.analog.com/en/resources/app-notes/an-922.html) This normalization facilitates straightforward [implementation](/page/Implementation) in transmitter and [receiver](/page/Receiver) chains without additional [scaling](/page/Scaling) factors. Graphically, $ h(t) $ exhibits a sinc-like [main lobe](/page/Main_lobe) centered [at $ t](/page/AT&T) = 0 $, with decaying oscillatory tails that [cross](/page/Cross) zero [at $ t](/page/AT&T) = \pm T, \pm 2T, \ldots $.[](https://www.analog.com/en/resources/app-notes/an-922.html) The cosine [term](/page/Term) modulated by $ \alpha $ smooths the response, reducing the ringing in the side lobes for higher values of $ \alpha $; for $ \alpha = 0 $, it reduces to a pure sinc function with more pronounced oscillations, while $ \alpha = 1 $ yields the smoothest decay.[](https://www.analog.com/en/resources/app-notes/an-922.html) ### Frequency Response The frequency response of the raised-cosine filter, denoted as $ H(f) $, is defined piecewise to ensure controlled spectral shaping while adhering to the [Nyquist criterion](/page/Nyquist_criterion) for zero [intersymbol interference](/page/Intersymbol_interference) ([ISI](/page/ISI)).[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) For frequencies satisfying $ |f| \leq \frac{1 - \alpha}{2T} $, the response is flat: $$ H(f) = T, $$ where $ T $ is the symbol period and $ \alpha $ (0 ≤ α ≤ 1) is the roll-off factor. In the transition region, $ \frac{1 - \alpha}{2T} < |f| \leq \frac{1 + \alpha}{2T} $, it follows a cosine-squared roll-off: $$ H(f) = \frac{T}{2} \left[ 1 + \cos\left( \frac{\pi T}{\alpha} \left( |f| - \frac{1 - \alpha}{2T} \right) \right) \right]. $$ Beyond this, for $ |f| > \frac{1 + \alpha}{2T} $, the response sharply drops to zero: $ H(f) = 0 $. This formulation originates from standard pulse-shaping theory in digital communications.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf)[](https://engineering.purdue.edu/~ee538/SquareRootRaisedCosine.pdf) The filter exhibits a flat [passband](/page/Passband) up to $ \frac{1 - \alpha}{2T} $, providing ideal [transmission](/page/Transmission) without [attenuation](/page/Attenuation) in the primary [bandwidth](/page/Bandwidth), followed by a smooth cosine-squared transition that minimizes ringing in the [time domain](/page/Time_domain). The sharp cutoff at $ \frac{1 + \alpha}{2T} $ confines the spectrum, preventing energy spillover into adjacent channels. The response maintains vestigial symmetry around the [Nyquist frequency](/page/Nyquist_frequency) $ \frac{1}{2T} $, where the transition band's "raised" portion ensures that the sum of shifted replicas $ \sum_{k=-\infty}^{\infty} H\left(f + \frac{k}{T}\right) = T $, satisfying the [Nyquist criterion](/page/Nyquist_criterion) for zero [ISI](/page/ISI) by creating equal ripple across [passband](/page/Passband) and [stopband](/page/Stopband) contributions.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf)[](https://www.analog.com/en/resources/app-notes/an-922.html) Graphically, the magnitude $ |H(f)| $ forms a characteristic raised-cosine curve symmetric about [DC](/page/DC), with the [passband](/page/Passband) as a rectangular plateau at height $ T $, sloping via the cosine term in the transition (width proportional to $ \alpha $), and nulling abruptly thereafter; increasing $ \alpha $ widens the raised portion, trading [bandwidth](/page/Bandwidth) for smoother [roll-off](/page/Roll-off) and reduced time-domain [sidelobes](/page/Sidelobes).[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) ### Roll-off Factor The roll-off factor, denoted as α (where 0 ≤ α ≤ 1), is a key design parameter in the [raised-cosine filter](/page/Raised-cosine_filter) that determines the excess [bandwidth](/page/Bandwidth), resulting in a total [bandwidth](/page/Bandwidth) of (1 + α)/(2T) compared to the minimum Nyquist [bandwidth](/page/Bandwidth) of 1/(2T), with T representing the symbol period.[](https://www.analog.com/en/resources/app-notes/an-922.html) When α = 0, the [filter](/page/Filter) reduces to an ideal [sinc function](/page/Sinc_function) in the [time domain](/page/Time_domain) with a rectangular [spectrum](/page/Spectrum), achieving the theoretical minimum [bandwidth](/page/Bandwidth) but at the cost of infinite duration.[](https://www.dsprelated.com/showarticle/60.php) At α = 1, the roll-off is maximized, resulting in a smoother transition band shaped as a half-cosine in the [frequency domain](/page/Frequency_domain), which doubles the occupied [bandwidth](/page/Bandwidth) relative to the Nyquist limit.[](https://www.analog.com/en/resources/app-notes/an-922.html) Higher values of α expand the transition bandwidth, thereby sacrificing spectral efficiency by increasing the overall signal bandwidth, while simultaneously reducing ringing artifacts in the time domain to enhance tolerance against intersymbol interference (ISI).[](https://www.dsprelated.com/showarticle/60.php) Conversely, lower α values constrain the bandwidth more tightly but amplify time-domain oscillations, exacerbating ISI in practical systems with timing errors.[](https://www.analog.com/en/resources/app-notes/an-922.html) A common choice is α = 0.5, which provides a balanced compromise between bandwidth conservation and ISI mitigation, widely adopted in standards like those for digital modulation schemes.[](https://www.mathworks.com/help/comm/ug/raised-cosine-filtering.html) For α = 0, the infinite temporal extent of the sinc [impulse response](/page/Impulse_response) leads to significant practical challenges, such as requiring [truncation](/page/Truncation) that introduces [approximation](/page/Approximation) errors and residual [ISI](/page/ISI).[](https://www.dsprelated.com/showarticle/60.php) In contrast, α = 1 yields a [frequency response](/page/Frequency_response) where the transition band follows a cosine curve from the [passband](/page/Passband) edge to full [stopband](/page/Stopband) attenuation, effectively doubling the [bandwidth](/page/Bandwidth) but minimizing overshoot and [sidelobes](/page/Sidelobes) in the eye [diagram](/page/Diagram), which aids in robust symbol detection.[](https://www.analog.com/en/resources/app-notes/an-922.html) The roll-off factor influences key performance trade-offs, including [bit error rate](/page/Bit_error_rate) (BER) versus [channel capacity](/page/Channel_capacity); lower α enhances [spectral efficiency](/page/Spectral_efficiency) and capacity by minimizing excess [bandwidth](/page/Bandwidth) but can elevate BER due to heightened [ISI](/page/ISI) sensitivity in noisy or dispersive channels, whereas higher α improves BER by smoothing the response and reducing ISI at the expense of reduced capacity from wider [bandwidth](/page/Bandwidth) usage.[](https://www.researchgate.net/publication/377527604_Finding_the_optimum_Roll-Off_rate_of_a_Raised_Cosine_Filter_at_the_Receiver_to_Achieve_the_Lowest_Bit_Error_Rate_for_a_given_Modulation_Order_and_a_given_EnergyBit_Normalized_Noise_Ratio) ## Key Properties ### Bandwidth Characteristics The bandwidth of a raised-cosine filter is fundamentally tied to the [Nyquist criterion](/page/Nyquist_criterion) for intersymbol interference-free transmission, where the minimum double-sided [bandwidth](/page/Bandwidth) required is the Nyquist [bandwidth](/page/Bandwidth) of $ \frac{1}{2T} $, with $ T $ denoting the symbol period. For a roll-off factor $ \alpha = 0 $, the filter achieves this exact Nyquist [bandwidth](/page/Bandwidth), corresponding to an ideal rectangular spectrum. However, practical implementations introduce $ \alpha > 0 $ to smooth the spectral transition, resulting in a total double-sided [bandwidth](/page/Bandwidth) $ B = \frac{1 + \alpha}{2T} $. This formulation ensures the filter's [frequency response](/page/Frequency_response) is strictly zero outside this band, confining all signal power within $ B $.[](https://www.analog.com/en/resources/app-notes/an-922.html) The excess bandwidth, defined as $ \alpha $ times the Nyquist bandwidth or $ \frac{\alpha}{2T} $, represents the additional spectral occupancy beyond the theoretical minimum, attributable to the "raised" cosine portion of the filter's [frequency response](/page/Frequency_response). This excess facilitates easier filter realization by reducing the sharpness of the cutoff but trades off [spectral efficiency](/page/Spectral_efficiency), as higher $ \alpha $ values increase overall [bandwidth](/page/Bandwidth) usage. In spectral efficiency analyses, the excess bandwidth quantifies the compromise between minimizing [intersymbol interference](/page/Intersymbol_interference) and optimizing channel utilization in band-limited systems.[](https://ntrs.nasa.gov/api/citations/20120008631/downloads/20120008631.pdf) Regarding power containment, the raised-cosine filter's design ensures that 100% of the signal power is contained within the [bandwidth](/page/Bandwidth) $ B = \frac{1 + \alpha}{2T} $, with [out-of-band](/page/Out-of-band) emissions zero due to the abrupt zeroing of the response beyond this limit. This property makes it ideal for meeting regulatory spectral masks in communications. For instance, with a common [roll-off](/page/Roll-off) factor of $ \alpha = 0.35 $, the bandwidth becomes approximately $ B \approx \frac{0.675}{T} $, striking a balance between efficiency and practical filtering constraints while maintaining total power confinement.[](https://www.analog.com/en/resources/app-notes/an-922.html) ### Autocorrelation Function The autocorrelation function of the raised-cosine filter impulse response $h(t)$ is defined as $$ R(\tau) = \int_{-\infty}^{\infty} h(t) h(t + \tau) \, dt, $$ representing the correlation between the pulse and its time-shifted version.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) By the Wiener-Khinchin theorem, $R(\tau)$ equals the inverse Fourier transform of the squared magnitude of the frequency response, $|H(f)|^2$, where $H(f)$ is the raised-cosine spectrum.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) This derivation links the time-domain correlation directly to the power spectral density, yielding a form similar to the raised-cosine shape but in the time domain. The explicit expression is $$ R(\tau) = T \left[ \frac{\sinc\left(\frac{\tau}{T}\right) \cos\left(\pi \alpha \frac{\tau}{T}\right)}{1 - \left(2 \alpha \frac{\tau}{T}\right)^2} - \frac{\alpha}{4} \sinc\left(\alpha \frac{\tau}{T}\right) \frac{\cos\left(\pi \frac{\tau}{T}\right)}{1 - \left(\alpha \frac{\tau}{T}\right)^2} \right], $$ with $\sinc(x) = \sin(\pi x)/(\pi x)$, $T$ the symbol period, and $\alpha$ the roll-off factor ($0 \leq \alpha \leq 1$). The second term resolves singularities at specific $\tau$ values, such as $\tau = T/(4\alpha)$.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf)[](https://engineering.purdue.edu/~ee538/SquareRootRaisedCosine.pdf) Key properties include a peak at $\tau = 0$ equal to the pulse energy (normalized to $T$) and zeros at $\tau = kT$ for nonzero integers $k$, ensuring orthogonality of symbol-spaced pulses.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) These traits make it suitable for [matched filter](/page/Matched_filter) receivers, where the filter matches the transmit [pulse](/page/Pulse) to produce $R(\tau)$ at the output.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) In additive white Gaussian noise (AWGN) channels, this configuration maximizes the signal-to-noise ratio (SNR) at sampling points, as the matched filter is optimal for detecting known signals in uncorrelated noise.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) The function exhibits a central lobe with a triangular-like envelope, decaying with sidelobes that cross zero at multiples of $T$, promoting clean symbol detection without intersymbol interference.[](https://daskalakispiros.com/files/Ebooks/digital-communication-proakis-salehi-5th-edition.pdf) ## Design and Implementation ### Parameter Selection The roll-off factor α, ranging from 0 to 1, is a critical parameter in raised-cosine filter design, balancing [spectral efficiency](/page/Spectral_efficiency) against robustness to imperfections such as timing errors. In bandwidth-constrained systems, a low α value, typically 0.2 to 0.4, is selected to limit excess bandwidth to (1 + α)/(2T), thereby maximizing the data rate within the available [spectrum](/page/Spectrum).[](https://complextoreal.com/wp-content/uploads/2013/01/isi.pdf) Higher α values, such as 0.5 to 1, are chosen for environments with [multipath propagation](/page/Multipath_propagation), where they enhance tolerance to timing recovery errors by widening the eye opening and reducing [intersymbol interference](/page/Intersymbol_interference) (ISI) sensitivity.[](https://dsp.stackexchange.com/questions/53213/confussion-on-roll-off-factor-sampling-time-and-raised-cosine-pulses) The symbol period T is directly determined by the required data rate R, with T = 1/R, ensuring the [filter](/page/Filter)'s bandwidth aligns with the channel's capacity to support the signaling rate without excessive [ISI](/page/ISI).[](https://www.mathworks.com/help/comm/ug/raised-cosine-filtering.html) In discrete-time implementations, the [filter](/page/Filter) length, or number of taps N, must be chosen for practical truncation of the [infinite impulse response](/page/Infinite_impulse_response) while minimizing [ISI](/page/ISI) from sidelobes. A common rule of thumb sets the span in symbols L_symbol ≈ 4 + 3/α for 0.2 < α < 0.75, with N = L_symbol × samples per symbol; for example, with α = 0.25 and 4 samples per symbol, this yields N ≈ 64 taps.[](https://engineering.purdue.edu/~ee538/SquareRootRaisedCosine.pdf) Additionally, the [filter](/page/Filter) coefficients are normalized to achieve unity gain, often by scaling so that the response at t = 0 equals 1 or the DC gain is unity, preserving signal amplitude through the transmit-receive chain.[](https://www.mathworks.com/help/signal/ref/rcosdesign.html) Parameter optimization involves simulations to evaluate trade-offs in ISI, eye diagram opening, and compliance with spectral masks, often guided by communication standards. For instance, IEEE 802.11g specifies a square-root raised-cosine filter with α = 0.25 to meet bandwidth and interference requirements in wireless LANs.[](https://mentor.ieee.org/802.11/dcn/01/11-01-0017-00-000g-functional-requirements-and-comparison-criteria-for-supergold-s-proposal-for-ieee-802-11g-high-rate-standard.doc) ### Practical Realization In digital systems, raised-cosine filters are typically realized as finite impulse response (FIR) filters, where the coefficients are obtained by sampling the continuous-time impulse response $ h(t) $ at the desired rate and truncating it to a finite length to make implementation feasible.[](https://www.analog.com/en/resources/app-notes/an-922.html) This truncation approximates the ideal infinite-duration response but introduces sidelobe ripples, which can be mitigated by applying a window function such as the [Hamming window](/page/Hamming_window) to reduce the Gibbs phenomenon near the transition band.[](https://www.mathworks.com/help/signal/ug/fir-filter-design.html) The filter length is often chosen as a multiple of the symbol period, with typical spans of 6 to 10 symbols for practical trade-offs between complexity and performance.[](https://www.dsprelated.com/showarticle/156.php) For enhanced noise rejection in communication systems, the root-raised cosine filter is commonly employed, with the filtering split equally between the transmitter and receiver to achieve matched filtering; the cascade of these square-root versions yields the overall raised-cosine response.[](https://www.mathworks.com/help/comm/ug/raised-cosine-filtering.html) This partitioning distributes the computational load and simplifies equalization at the receiver.[](https://www.gaussianwaves.com/2018/10/square-root-raised-cosine-pulse-shaping/) Filter coefficients can be generated in software environments like MATLAB using built-in functions such as `rcosdesign`, which computes the response via direct sinc-based formulas, or through frequency-domain methods involving [FFT](/page/FFT)/[IFFT](/page/IFFT) for efficient design with complexity $ O(N \log N) $, where $ N $ is the filter length.[](https://www.mathworks.com/help/signal/ref/rcosdesign.html) In Python, implementations often rely on libraries like [NumPy](/page/NumPy) and [SciPy](/page/SciPy) for sinc computations or inverse Fourier transforms to derive the coefficients, enabling rapid prototyping without specialized hardware.[](https://pysdr.org/content/pulse_shaping.html) These tools facilitate parameter tuning based on roll-off factor and oversampling rate selections from prior design stages. A key challenge in realization is approximating the infinite-duration impulse response through truncation, which can cause spectral aliasing if not addressed by oversampling the signal at 4 to 8 samples per symbol during filter design and application.[](https://wirelesspi.com/pulse-shaping-filter/) This oversampling ensures sufficient padding in the time domain to suppress aliasing artifacts while maintaining the filter's intersymbol interference minimization properties.[](https://www.analog.com/en/resources/app-notes/an-922.html) Hardware implementations, such as in FPGAs or DSP chips, further consider multiplierless approximations using distributed arithmetic to reduce resource usage without significant performance degradation.[](https://www.iosrjournals.org/iosr-jvlsi/papers/vol7-issue5/Version-1/K0705016973.pdf) ## Applications ### In Digital Communications In digital communications, the raised-cosine filter serves as a fundamental pulse-shaping tool at the transmitter, where it band-limits transmitted symbols to control the signal spectrum and prevent spectral regrowth, particularly in modulation schemes like [phase-shift keying (PSK)](/page/Phase-shift_keying) and [quadrature amplitude modulation (QAM)](/page/Quadrature_amplitude_modulation).[](https://ieeexplore.ieee.org/document/1169389/) This band-limiting ensures compliance with spectral masks while minimizing out-of-band emissions, thereby enhancing overall system efficiency in bandwidth-constrained environments.[](https://ieeexplore.ieee.org/document/8097061/) At the receiver, a root-raised-cosine filter implements matched filtering, which optimally restores the transmitted pulse shape and maximizes the signal-to-noise ratio (SNR) by correlating the received signal with the known transmit pulse; when paired with the transmitter's root-raised-cosine filter, the overall channel response forms a full [raised-cosine filter](/page/Raised-cosine_filter) that eliminates intersymbol interference (ISI) at sampling instants.[](https://ieeexplore.ieee.org/document/634674/) The autocorrelation properties of the raised-cosine pulse further support this SNR maximization by concentrating signal energy at the decision point.[](https://ieeexplore.ieee.org/document/9415511/) Raised-cosine filters are integral to several communication standards, including [LTE](/page/LTE) and [5G NR](/page/5G_NR) where a roll-off factor β of 0.22 is commonly specified for root-raised-cosine pulse shaping in baseband processing,[](https://www.etsi.org/deliver/etsi_ts/138100_138199/13810101/15.08.02_60/ts_13810101v150802p.pdf) as well as [DVB-T](/page/DVB-T) with β=0.15[](https://www.etsi.org/deliver/etsi_en/300700_300799/300749/01.01.02_60/en_300749v010102p.pdf) and cable modems under [DOCSIS](/page/DOCSIS) specifications using β=0.25.[](https://account.cablelabs.com/server/alfresco/1cee0cc2-02f1-47e5-8ec8-a7c2ed8dd8ae) In these systems, eye diagram analysis reveals improved eye opening due to reduced ISI, facilitating reliable symbol detection even in multipath channels.[](https://ieeexplore.ieee.org/document/7209079/) By confining the signal spectrum within the allocated bandwidth, raised-cosine filters effectively reduce adjacent channel interference, allowing denser frequency reuse in cellular and broadcast networks.[](https://ieeexplore.ieee.org/document/7993492/) For instance, in 16-QAM systems employing a roll-off factor β=0.3, bit error rates below 10^{-5} are achievable at an SNR of 20 dB, demonstrating robust performance under typical noise conditions.[](https://ieeexplore.ieee.org/document/7209079/) ### Related Uses and Variants The raised-cosine filter has been adapted in quasi-constant envelope modulation schemes, such as [staggered quadrature overlapped raised cosine](/page/staggered_quadrature_overlapped_raised_cosine) (SQORC) and [filtered quadrature phase-shift keying](/page/filtered_quadrature_phase-shift_keying) (FQPSK), to approximate constant envelope properties similar to [Gaussian minimum shift keying](/page/Gaussian_minimum_shift_keying) (GMSK) while improving spectral efficiency. In SQORC, the pulse shape incorporates a raised-cosine response to limit envelope fluctuations to about 3 dB, enabling higher data rates than GMSK without fully sacrificing power amplifier efficiency, though additional filtering in FQPSK-B variants is needed to match GMSK's out-of-band spectral containment at BT_b = 0.5.[](https://descanso.jpl.nasa.gov/monograph/series3/chapter3.pdf) In optical communications, raised-cosine filters are used for compensating chromatic and intermodal dispersion in fiber links, mitigating inter-symbol interference by shaping pulses to align delayed signal components. Simulations demonstrate that these filters enhance eye diagram quality and bit error rates in wavelength-division multiplexing systems, offering lower implementation complexity than traditional all-pass filters due to their Nyquist properties and tunable roll-off factor β.[](https://www.researchgate.net/publication/265206809_Dispersion_Compensation_Using_Raised_Cosine_Filter_inOptical_Fibers) Beyond communications, raised-cosine profiles are employed in audio signal processing for smooth crossfades and fading effects, where their S-curve shape ensures constant power transitions between segments, avoiding audible clicks or dips. This power-complementary behavior, inherent to the filter's cosine-based taper, is particularly useful in music synthesis and editing software for seamless overlaps.[](https://sethares.engr.wisc.edu/paperspdf/audiomorphing.pdf) In image processing, truncated raised-cosine pulses function as interpolation kernels for resizing, providing smoother frequency roll-off than ideal sinc functions to suppress Gibbs ringing artifacts at edges while preserving sharpness. Comparative evaluations show reduced mean squared error in resampled images, with the filter's finite support enabling efficient computation for real-time applications.[](https://ieeexplore.ieee.org/document/1439123) Compared to the sinc filter, which corresponds to a raised-cosine with roll-off factor β = 0 and offers the sharpest bandwidth transition for zero ISI, the raised-cosine variant (β > 0) is more practical due to its finite-duration approximation, which mitigates excessive ringing from the sinc's infinite tails and [Gibbs phenomenon](/page/Gibbs_phenomenon) in time-limited implementations.[](https://www.analog.com/en/resources/app-notes/an-922.html) In contrast, the Blackman-Harris window achieves superior sidelobe suppression (up to -92 [dB](/page/DB) peak) over raised-cosine-based windows like the Hann (-31 [dB](/page/DB)), making it preferable for [spectral analysis](/page/Spectral_analysis) requiring minimal leakage, though at the cost of higher design complexity from its four-term formulation.[](https://www.cs.cmu.edu/afs/cs/user/bhiksha/WWW/courses/dsp/spring2013/WWW/schedule/readings/windows_comparison2_harris.pdf) Adaptive raised-cosine filters with dynamically adjusted [roll-off](/page/Roll-off) factors have been explored in filter-bank multicarrier schemes like FBMC to optimize performance in dispersive channels.[](http://wcsp.eng.usf.edu/courses/wcsl/documents/publications/Esad_SDR_2013.pdf) Emerging research as of 2025 investigates raised-cosine variants in [6G](/page/6G) systems for integrated sensing and communication (ISAC), where tunable [roll-off](/page/Roll-off) enhances joint radar-communication efficiency.[](https://ieeexplore.ieee.org/document/10456789)

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