CS 536 Park FDM/OFDM and Related Concepts - Flashcards

FDM, FDMA, and OFDM: Core Concepts

  • Key idea: send bits over multiple carrier frequencies using sinusoids; frequency is the primary carrier variable.

    • A carrier frequency f with a sinusoid, e.g., s(t) ≈ A(t) sin(f t), encodes bits by changing amplitude or presence of the tone.
    • Low amplitude can represent 0; high amplitude can represent 1; or turning the carrier on/off encodes bits.
    • Carrier frequency f is the “DNA” of the signal: the spectral content determines how bits are spread in frequency.

  • Transmission of multiple streams in parallel:

    • Use multiple carrier frequencies f1, f2, …, f_n; this divides the available spectrum among users or bit streams.
    • This approach is called Frequency Division Multiplexing (FDM).
    • In optical fiber, the term Wave Division Multiplexing (WDM) is used.
    • Two primary application scenarios:
    • Multi-user: FDMA — each user gets a distinct frequency band.
    • Single-user: all frequencies are used by one user to ship bits in parallel, reducing completion time for a group of bits.

  • Relevance to real-world wireless: spread spectrum and robustness concepts

    • Confidentiality and anti-eavesdropping: hopping across frequencies (frequency hopping) makes interception harder because transmission is effectively sequential over frequencies.
    • Anti-jamming: spreading bits over multiple frequencies makes targeted jamming harder.
    • Frequency-selective fading: some frequencies distort more than others; spreading across frequencies and using error correction improves reliability.
    • Referred to as spread spectrum; bandwidth here means the frequency range from f1 to fn over which the bits are spread. Example: f1 = 1 GHz, f{10} = 1.9 GHz → bandwidth = 0.9 GHz.

  • Fundamental mathematical framework: complex sinusoids and Euler’s formula

    • Complex sinusoid: e^{i f t} =
      \cos(f t) + i \sin(f t)
    • Euler’s formula connects real sinusoids to complex exponentials; useful for linear-algebra and Fourier-domain analyses.
    • This perspective underpins CDMA-style analysis and synthesis (see below).

  • Linear-algebra view of CDMA (code-based multiple access)

    • Finite dimension n: number of users; each user sends a single bit per symbol interval.
    • Fix an orthogonal basis x1, x2, …, x_n (code vectors) in an n-dimensional vector space.
    • Any vector z can be expressed as a weighted sum: z = \sum{k=1}^n ak x_k.
    • Encoding (synthesis): to send n bits, hide the n bits in the scalar weights a_k. The vector z encodes the message.
    • Decoding (analysis): for user i, compute z ◦ xi (inner product) to obtain ai. Orthogonality ensures ai is extracted without interference from other ak.
    • Implication: CDMA can be viewed as projecting the message onto an orthogonal code basis to separate users.

  • Continuous-frequency FDMA view: inverse Fourier perspective

    • In FDMA with complex sinusoids, the space is infinite-dimensional (time is unbounded and continuous).
    • Basis elements: complex sinusoids e^{i f t} for different frequencies f.
    • The signal of interest is a weighted sum: s(t) = \frac{1}{2\pi} \int{-\infty}^{\infty} af e^{i f t} df.
    • Synthesis: hidden bit information in the weights af. If the signal is composed of a finite number of carrier frequencies, only a few af are nonzero.
    • This is the Fourier synthesis viewpoint; the corresponding analysis view yields the Fourier transform.

  • Fourier analysis: synthesis and analysis basics

    • Given signal s(t), the Fourier coefficient at frequency f is af = \int{-\infty}^{\infty} s(t) e^{-i f t} dt.
    • In shorthand, af is the Fourier transform of s(t) at frequency f: af = \mathcal{F}
      {s}(f).
    • In practice, fast algorithms like the Fast Fourier Transform (FFT) compute these coefficients efficiently; synthesis uses the inverse FFT (IFFT).
    • Conceptual mapping: discrete, finite sums correspond to a finite set of coefficients; continuous sums correspond to a continuous spectrum.

  • Why we care about bandlimited signals and bandwidth

    • Bandwidth is the range of frequencies needed to represent s(t): bandwidth B = fb - fa.
    • If s(t) is bandlimited to [fa, fb], only frequencies within this range carry energy and information.
    • The term bandwidth is sometimes conflated with data rate (bps) or with the transmitter/medium’s operational range; in practice, bandlimited signals are approximated when necessary.
    • Practical signals are often approximated as bandlimited by neglecting negligible spectral components.

  • Illustrative spectra and encoding examples

    • Example 1: s(t) = sin(10 t) + 3 sin(20 t) + sin(30 t)
    • Spectrum shows components at 10 Hz (amplitude 1), 20 Hz (amplitude 3), 30 Hz (amplitude 1).
    • Example 2: s(t) = 0.1 sin(1 t) + sin(2 t) + 0.1 sin(3 t)
    • Spectrum concentrated at 1 Hz, 2 Hz, 3 Hz with respective amplitudes.
    • Example 3: s(t) = sin(2 t) (single sine)
    • Admits a single spectral component at 2 Hz; other frequency components ideally zero.
    • Example: square wave spectrum
    • A square wave has an infinite spectrum (sinc-like decay of coefficients). In practice, we truncate small coefficients and approximate.

  • Inter-channel interference (ICI) in traditional FDM

    • Modulating one bit stream on carrier f = 100 MHz spreads energy into nearby frequencies, creating a spectral “bump.”
    • If another stream occupies nearby carrier f = 101 MHz, the bumps may overlap, causing ICI and decoding difficulty.
    • Good case: sufficiently separated carriers (guard bands) prevent overlap; bad case: significant overlap leads to superposition and decoding error.
    • Guard bands are thus essential in traditional FDM to minimize ICI.
    • Drawback: guard bands reduce spectral efficiency because portions of the spectrum are unused.

  • Regulatory and practical spectrum considerations

    • The physical medium has a finite bandwidth; outside this bandwidth, distortion is too high.
    • Wireless spectrum is crowded; spectral efficiency (bits per second per Hz) is a premium resource.
    • Regulation and sharing constraints influence how many carriers can be packed into a given band.

  • Orthogonal Frequency Division Multiplexing (OFDM) vs traditional FDM

    • OFDM uses a set of sinusoids that are mutually orthogonal over a finite time window τ.
    • Mathematical orthogonality condition:
      \int{-\tau/2}^{\tau/2} e^{i fi t} e^{-i f_j t} \, dt = 0 \quad \text{for } i \neq j.
    • Finite support: orthogonality is achieved over a chosen symbol interval [-τ/2, τ/2]. The interval length τ is analogous to the baud (symbol) period.
    • Advantage: multiple carrier frequencies can spectrally overlap without causing ICI because they are orthogonal within the interval τ.
    • Practical implication: OFDM enables higher spectral efficiency by allowing dense packing of carriers with overlapping spectra.

  • How to select mutually orthogonal carriers within a given bandwidth

    • Target band: [fa, fb] with bandwidth W = fb - fa.
    • Choose n carrier frequencies as:
      f1 = fa + \Delta f, \; f2 = fa + 2\Delta f, \; \dots, \; fn = fa + n\Delta f,
      where the spacing is
      \Delta f = \frac{W}{n}.
    • The symbol period is then
      \tau = \frac{n}{W}.
    • Trade-off relationships:
    • Increasing n (more carriers) increases spectral efficiency but lengthens the symbol period, reducing per-user bps if W is fixed.
    • Increasing W (more total bandwidth) reduces the symbol period, increasing potential bps but requiring more spectrum.

  • Bandwidth vs bit rate: a conservation principle

    • For n parallel bit streams with two-level signaling (AM with 2 levels), each carrier transmits 1/τ bps.
    • Total bps with n carriers: n/τ.
    • Since τ = n/W, total bps equals W (independent of n):
      ext{Total bps} = n \cdot \frac{1}{\tau} = \frac{n}{\tau} = W.
    • This is the essence of Shannon’s second theorem: capacity scales with bandwidth; adding more carriers within the same bandwidth does not increase total capacity beyond W, once noise is accounted for.
    • Practical note: noise and modulation schemes affect achievable rates beyond this idealized conservation.

  • Practical examples and implications

    • Example: Available bandwidth from 2.4 GHz to 2.5 GHz (W = 100 MHz), with n = 100 carriers
    • Carrier spacing: \Delta f = W/n = 1 \text{ MHz}.
    • Carrier frequencies: 2.401 GHz, 2.402 GHz, …, 2.5 GHz.
    • Symbol period: τ = n/W = 100 / (100 \times 10^6) = 1 \mu s.
    • AM with 2 levels: 1 Mbps per user; total capacity: 100 Mbps.
    • General takeaway: the data rate is determined by bandwidth W, not the absolute carrier frequency f (e.g., W = 100 MHz yields similar bps across bands like 2.4 GHz vs 5 GHz, or different absolute carrier centers).
    • OFDM processing typically happens at a time granularity of about τ (e.g., 1 μs in the example); the actual RF carrier upshift to the desired band is a separate translation step (e.g., modulating the 1 MHz baseband onto a 2.4 GHz carrier).

  • Wireless network examples and access methods

    • IEEE 802.11g (2.4 GHz band)
    • OFDM is used with W = 20 MHz and n = 64 carriers.
    • Carrier spacing: 20 MHz / 64 = 312.5 kHz.
    • Symbol time: τ = 64 / 20e6 = 3.2 μs.
    • Important caveat: OFDM in 802.11g is not OFDMA; it uses a single user at a time per channel; multiple access is achieved via higher-layer protocols (CSMA/CA).
    • Similar approach for 802.11a/n in the 5 GHz band. 802.11ax (Wi-Fi 6) introduces OFDMA (multi-user OFDM) to share subcarriers among users; Wi-Fi 7 (802.11be) ratified in 2024.
    • ADSL example (copper telephone line): ITU G.992.1; W = 1.104 MHz, n = 256 → carrier spacing ≈ 4.3125 kHz; uses OFDM, not OFDMA. OFDMA is considered for optical fiber networks (extends WDM).

  • Summary of key takeaways

    • FDM distributes data across non-overlapping or orthogonal frequency bands; guard bands mitigate ICI but reduce efficiency.
    • OFDM achieves high spectral efficiency by packing many carriers into a bandwidth with orthogonality over the symbol interval τ.
    • The spectral content of a signal (Fourier transform) reveals how information is distributed across frequencies; synthesis and analysis are dual views (weights a_f vs time-domain signal).
    • Real-world constraints include spectrum regulation, regulatory limits, and practical concerns about noise and interference; optimal designs balance n, W, τ to meet performance goals.

  • Key equations to remember

    • Synthesis with finite carriers:
      s(t) = \sum{k=1}^{n} ak e^{i f_k t}
    • Continuous Fourier synthesis:
      s(t) = \frac{1}{2\pi} \int{-\infty}^{\infty} af e^{i f t} \mathrm{d}f
    • Fourier coefficient (analysis):
      af = \int{-\infty}^{\infty} s(t) e^{-i f t} \mathrm{d}t
    • Orthogonality condition over finite interval:
      \int{-\tau/2}^{\tau/2} e^{i fi t} e^{-i f_j t} \mathrm{d}t = 0 \quad (i \neq j)
    • Carrier spacing and symbol period for finite-time orthogonal carriers:
      \Delta f = \frac{W}{n}, \quad \tau = \frac{n}{W}
    • Total capacity under the bandwidth constraint (conservation):
      \text{Total bps} = \frac{n}{\tau} = W

  • Notes on interpretation and pedagogy

    • Think of a signal’s spectrum as its DNA: the Fourier coefficients a_f tell you which frequencies matter and by how much.
    • Shifting from a purely synthesis view (how to build s(t) from basis functions) to an analysis view (how to recover a_f from s(t)) highlights how modulation schemes spread energy across the spectrum.
    • The trade-offs among n (number of carriers), W (available bandwidth), and τ (symbol period) are central to designing multipoint transmission systems and understanding why modern systems leverage OFDM rather than naive, wide-guard-band FDMA.