Signals and Fourier Analysis

Nyquist rate and sampling theorem

A low-pass signal whose highest frequency component is f_m Hz can be reconstructed from its samples only if the sampling frequency satisfies f_s \ge 2f_m. The minimum rate 2f_m is the Nyquist rate, and in rad/s it is 2\omega_m. When a new signal is derived from old ones, first find its new highest frequency and only then double it. Multiplication in time convolves the spectra in frequency, so the bandwidths of the factors add: if x_1 is band-limited to B_1 and x_2 to B_2, the product is band-limited to B_1+B_2, and its Nyquist rate is 2(B_1+B_2). Time scaling x(at) with |a|>1 compresses the signal and multiplies its bandwidth by |a|, while a time shift only adds linear phase and leaves the bandwidth unchanged. When a band-limited signal of bandwidth B multiplies a carrier \cos\omega_c t, its spectrum moves to \pm\omega_c and its highest frequency becomes \omega_c+B. For a sum of such modulated terms, the highest frequency is the largest of these values, and the Nyquist rate is twice it. In impulse-train sampling with an offset \tau, each replica of the sampled spectrum carries the phase factor e^{-jk\omega_s\tau}, and the k=0 replica carries no phase. If \omega_s>2\omega_c, i.e. T_s<\pi/\omega_c, no replicas overlap in the passband, so an ideal lowpass filter with passband gain T_s recovers f(t) itself, with no delay.

Exponential Fourier series coefficients

The exponential Fourier series represents a periodic signal as x(t)=\sum_k a_k e^{jk\omega_0 t}, where \omega_0=2\pi/T, and the coefficients are a_k=\frac{1}{T}\int_T x(t)e^{-jk\omega_0 t}\,dt. The coefficient a_0 is the average value of x(t). If x(t) is real then a_{-k}=a_k^*, and if it is also even then every a_k is real and a_{-k}=a_k. Parseval's relation for a periodic signal gives the average power as \sum_k |a_k|^2, so the coefficients can be used to compute power directly. Time scaling changes the period but not the coefficients. If y(t)=f(\alpha t) with \alpha>1, the exponentials of y are e^{jk(2\pi/T_0)\alpha t}, so d_k=c_k, and the fundamental period of y is T_0/\alpha, not \alpha T_0. Half-wave antisymmetry, x(t-T/2)=-x(t), forces the even-index coefficients to vanish. Shifting the integration variable by T/2 gives a_n=(-1)^{n+1}a_n, because e^{-jn\pi}=(-1)^n, so a_n=0 whenever n is even, and only odd harmonics remain.

Continuous-time convolution integral

The convolution y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau is evaluated by flipping one signal, shifting it by t, and integrating the overlap. The integration limits change whenever an edge of one signal crosses an edge of the other, so the result should be found interval by interval. When a finite-duration pulse is convolved with a decaying exponential, the overlap eventually covers the whole pulse, and for large t the result is a constant multiple of e^{-t}, which tends to zero as t\to\infty. For the total area, the convolution theorem at zero frequency gives \int y(t)\,dt=\left(\int x_1(t)\,dt\right)\left(\int x_2(t)\,dt\right). Each impulse contributes its weight and each rectangular pulse contributes its height times its width, so this shortcut avoids computing y(t) explicitly.

Spectra of periodic and aperiodic signals

Duality links periodicity in one domain to discreteness in the other. A continuous aperiodic signal has a continuous aperiodic spectrum. A continuous periodic signal has a discrete aperiodic spectrum, which is the line spectrum of a Fourier series. A discrete aperiodic signal has a continuous periodic spectrum, as in the DTFT. A discrete periodic signal has a discrete periodic spectrum, as in the DFS. Each signal type must be matched with the spectral type that follows this pattern. Sinusoids whose frequencies are integer multiples of a common fundamental are orthogonal over one fundamental period, meaning their inner product is zero. Orthonormal additionally requires each function to have unit energy, which sinusoids generally do not have over one period. A sinusoid whose frequency is an integer multiple of another's is called a harmonic of it. Over a partial period the inner product is in general nonzero, so orthogonality depends on the interval chosen.

Fourier transform of exponential signals

The basic pairs are e^{-at}u(t)\leftrightarrow\frac{1}{a+j\omega} for a>0 and e^{-a|t|}\leftrightarrow\frac{2a}{a^2+\omega^2}. By duality, the Lorentzian \frac{1}{1+t^2} transforms to \pi e^{-|\omega|}. The differentiation property \mathcal{F}\{\frac{d}{dt}f\}=j\omega F(\omega) extends these pairs to signals that are derivatives of them. A signal such as \frac{t}{(1+t^2)^2} equals -\frac{1}{2}\frac{d}{dt}\left[\frac{1}{1+t^2}\right], so its transform is -\frac{1}{2}j\omega\pi e^{-|\omega|}, which equals \frac{\pi}{2j}\omega e^{-|\omega|} because -j=1/j. An odd real time function has a purely imaginary and odd spectrum, so the factor j must be tracked carefully.

Distortionless LTI system and linear phase

An LTI system is distortionless when its frequency response has the form H(\omega)=K e^{-j\omega t_d}, that is, constant magnitude over the band of interest together with a phase that is linear in \omega. Its output is then y(t)=Kx(t-t_d). Magnitude alone controls amplitude distortion but says nothing about phase, so a system whose |H| equals one over a band is not necessarily a pure delay unless its phase is linear with slope -t_d. For a wide-sense stationary input with power spectral density S_X(f), the output PSD is S_Y(f)=|H(f)|^2S_X(f), which depends only on the magnitude response. The output PSD therefore equals the input PSD wherever |H(f)|=1, whatever the phase. Power spectra ignore phase, but waveforms do not.

LTI system linearity and time invariance

Linearity means superposition: a weighted sum of inputs produces the same weighted sum of outputs. Time invariance means that a delay of the input produces the same delay of the output. A key consequence for LTI systems is that when a sinusoid of frequency \omega_0 is applied, the steady-state output contains only that frequency, scaled and phase-shifted. Any output containing a frequency absent from the input, or a DC component when the input has none, cannot come from an LTI system. Nonlinear operations such as squaring create new frequencies: the square of a sinusoid contains a constant term and a term at twice the frequency, so a squaring system is not LTI.