Signals and Systems

This note is a revised and compressed version of Signals and Systems material prepared around 2015. The goal is not formula memorization but one coherent view of time, frequency, and the way systems modify signals.

Signals and systems

A signal carries information as a function of one or more independent variables; continuous-time signals are written x(t) and discrete-time signals x[n]. A system maps an input to an output:

x -> system -> y

A system may be memoryless, causal, BIBO stable, time invariant, linear, or invertible. Linearity requires both additivity and homogeneity; y=2x+3 is not linear, while y[n]=x[n-1] is causal but has memory.

Elementary signals

In discrete time:

δ[n] = u[n] - u[n-1]
u[n] = Σ(k=-∞..n) δ[k]

In continuous time the Dirac impulse is defined by its action under integration:

u(t) = ∫(-∞..t)δ(τ)dτ
δ(t) = du(t)/dt
∫x(t)δ(t-t0)dt = x(t0)

The complex exponential,

e^(jωt) = cos(ωt) + j sin(ωt)

is the natural basis for sinusoidal analysis. Continuous-time e^(jω0t) is periodic for every nonzero ω0; discrete-time e^(jω0n) is periodic only when ω0/(2π) is rational, and discrete frequency itself is 2π periodic.

Energy:

E = ∫|x(t)|²dt
E = Σ|x[n]|²

classifies finite-energy signals; periodic signals instead have infinite energy but finite nonzero average power.

Time transformations

x(t-t0)   shift
x(-t)     reversal
x(at)     scaling

Continuous-time scaling is arbitrary; discrete-time indices must remain integral, so rate change is treated as sample removal or insertion.

Even and odd parts are:

x_e(t)=[x(t)+x(-t)]/2
x_o(t)=[x(t)-x(-t)]/2

LTI systems and convolution

A linear time-invariant system is completely determined by its impulse response.

Discrete time:

x[n]=Σx[k]δ[n-k]
y[n]=Σx[k]h[n-k]=x[n]*h[n]

Continuous time:

y(t)=∫x(τ)h(t-τ)dτ=x(t)*h(t)

Convolution is commutative, associative, and distributive; cascaded LTI systems convolve impulse responses and parallel systems add them.

System properties follow from h:

memoryless   h(t)=Kδ(t)
causal       h(t)=0, t<0
stable       ∫|h(t)|dt < ∞

with Σ|h[n]|<∞ in discrete time.

Step and impulse responses satisfy:

s(t)=u(t)*h(t)
h(t)=ds(t)/dt
h[n]=s[n]-s[n-1]

Differential and difference equations

A continuous LTI system can be written:

Σa_k d^k y(t)/dt^k = Σb_k d^k x(t)/dt^k

and a discrete system:

Σa_k y[n-k] = Σb_k x[n-k]

Initial conditions determine the natural response. Recursive discrete systems generally produce IIR behavior; finite nonrecursive sums produce FIR behavior.

Eigenfunctions and frequency response

Complex exponentials are LTI eigenfunctions:

e^(st) -> H(s)e^(st)
z^n    -> H(z)z^n

The system preserves their form and changes only a scalar coefficient, so H(jω) directly describes sinusoidal amplitude and phase change.

Fourier series

For period T0 and ω0=2π/T0:

x(t)=Σa_k e^(jkω0t)
a_k=(1/T0)∫(T0)x(t)e^(-jkω0t)dt

a0 is the average value and real signals have conjugate-symmetric coefficients. Under the Dirichlet conditions, the series converges to the signal at continuity points and to the average of one-sided limits at jumps.

Truncation leaves an overshoot of roughly nine percent near a discontinuity; increasing the number of terms narrows but does not remove it. This is the Gibbs phenomenon.

Parseval:

(1/T0)∫|x(t)|²dt = Σ|a_k|²

connects time-domain average power to harmonic power. A periodic discrete-time sequence has only N distinct harmonics.

Fourier transform

For an aperiodic signal:

X(jω)=∫x(t)e^(-jωt)dt
x(t)=(1/2π)∫X(jω)e^(jωt)dω

Core properties are:

x(t-t0)     <-> e^(-jωt0)X(jω)
dx/dt       <-> jωX(jω)
x(at)       <-> (1/|a|)X(jω/a)
x*h         <-> XH
xh          <-> (1/2π)X*H

Compression in time expands frequency. Perfect time limitation and perfect band limitation cannot hold simultaneously.

Useful pairs include:

δ(t)          <-> 1
1             <-> 2πδ(ω)
e^(-at)u(t)   <-> 1/(a+jω)
rectangle     <-> sinc

Convolution becoming multiplication is the central simplification of LTI analysis.

DTFT, DFT, and FFT

The DTFT:

X(e^(jω))=Σx[n]e^(-jωn)
x[n]=(1/2π)∫(2π)X(e^(jω))e^(jωn)dω

is 2π periodic.

For N finite samples, the DFT is:

X[k]=Σ(n=0..N-1)x[n]e^(-j2πkn/N)
x[n]=(1/N)Σ(k=0..N-1)X[k]e^(j2πkn/N)

The DFT treats the block as periodic; endpoint mismatch produces spectral leakage.

The FFT is not another transform but a family of efficient DFT algorithms. Direct DFT costs about N² operations, while FFT methods require about N log N.

Windowing

Finite observation multiplies a signal by a window. A rectangular window gives a narrow main lobe but high sidelobes; Hann and Hamming reduce leakage, while Blackman suppresses sidelobes further at the cost of a wider main lobe.

Zero padding samples the displayed spectrum more densely but does not create true frequency resolution; observation duration remains the main limit.

STFT

A single Fourier transform gives frequency but not time. The short-time Fourier transform analyzes overlapping windows:

STFT -> spectrogram

Short windows improve time resolution and long windows improve frequency resolution. Speech, music, vibration, radar, and sonar analysis all expose this tradeoff.

Correlation and power spectrum

Cross-correlation,

r_xy[k]=Σx[n]y*[n-k]

measures similarity versus delay; autocorrelation compares a signal with shifted copies of itself. Delay estimation, periodicity detection, matched filtering, and synchronization use the same operation.

The Wiener-Khinchin relation connects autocorrelation and power spectral density. A finite-record periodogram has high variance; Welch's method averages windowed overlapping segments to reduce variance at the cost of frequency resolution.

Sampling and quantization

Impulse-train sampling:

x_p(t)=x(t)Σδ(t-nT)

replicates the spectrum at ωs=2π/T intervals:

X_p(jω)=(1/T)ΣX(j(ω-kωs))

For bandlimit ωM, the condition ωs>2ωM keeps replicas separate. 2ωM is the Nyquist rate, not the selected physical sampling rate.

Aliasing makes high-frequency components appear at lower frequencies and destroys information at sampling time. A practical ADC chain is therefore:

analog -> anti-alias filter -> sample/hold -> ADC -> digital data

Sampling discretizes time; quantization discretizes amplitude. An ideal B-bit ADC has 2^B levels and roughly:

SNR ≈ 6.02B + 1.76 dB

for a full-scale sinusoid. Thermal noise, clock jitter, and nonlinearity reduce practical ENOB.

Ideal reconstruction uses sinc interpolation; practical DACs use a zero-order hold followed by an analog reconstruction filter.

Multirate processing

Downsampling requires:

low-pass -> keep every Mth sample

and interpolation requires:

insert L-1 zeros -> low-pass

Without prefiltering, decimation aliases the discrete spectrum. Rational L/M conversion combines both operations; polyphase structures avoid computations that would later be discarded.

Filters

The main responses are low-pass, high-pass, band-pass, band-stop, and all-pass. A brick-wall response is noncausal, so real filters trade transition width, ripple, attenuation, phase, order, and computation.

FIR:

y[n]=Σb_k x[n-k]

is finite, nonrecursive, always BIBO stable, and can have exact linear phase.

IIR:

y[n]=-Σa_k y[n-k]+Σb_k x[n-k]

can reach a given selectivity with fewer coefficients but requires pole-stability and numerical-sensitivity analysis.

Classical families include Butterworth, Chebyshev I and II, elliptic, and Bessel filters, each choosing a different balance among flatness, transition sharpness, and phase behavior.

Phase and group delay

Distortionless transmission requires:

H(jω)=Ke^(-jωt0)

so magnitude is constant and phase is linear.

Group delay:

τg(ω)=-d∠H(jω)/dω

describes narrowband envelope delay. Frequency-dependent group delay spreads pulses and may distort communication symbols.

Bode magnitude uses:

20log10|H(jω)|

so cascade gains become sums in dB.

Laplace transform

X(s)=∫x(t)e^(-st)dt
s=σ+jω

extends Fourier analysis with exponential weighting and a region of convergence. The ROC is part of the transform because the same algebraic expression can represent different time-domain signals under different ROCs.

For a causal rational system, the ROC lies to the right of the rightmost pole; stability requires the jω axis to lie inside the ROC. A causal stable continuous-time rational system therefore has all poles in the left half-plane.

The system function:

H(s)=Y(s)/X(s)

comes directly from the differential equation under initial rest. The unilateral transform includes initial conditions:

L{dx/dt}=sX(s)-x(0-)

z transform

The discrete-time counterpart is:

X(z)=Σx[n]z^(-n)

and the unit circle corresponds to the DTFT.

For a causal rational discrete-time system, the ROC lies outside the outermost pole; stability requires the unit circle to lie inside the ROC. A causal stable system therefore has all poles inside the unit circle.

Difference equations produce:

H(z)=B(z)/A(z)

and poles near the unit circle create longer memory and sharper frequency selectivity.

Numerical realization

A high-order IIR implemented as one polynomial is numerically fragile; cascaded second-order sections are usually more robust:

H(z)=H1(z)H2(z)...Hm(z)

Overflow, rounding, coefficient quantization, and limit cycles are system effects in finite-precision implementations. Direct, cascade, and parallel realizations of the same theoretical transfer function can behave differently.

State space

A transfer function describes input-output behavior; state space exposes internal memory.

Continuous time:

x_dot=Ax+Bu
y=Cx+Du

Discrete time:

x[n+1]=Ax[n]+Bu[n]
y[n]=Cx[n]+Du[n]

The state is the smallest internal information set needed to predict future behavior given future input. MIMO systems, controllability, and observability are naturally expressed in this form.

Modulation and communications

Amplitude modulation follows:

x(t)cos(ωct)
<->
(1/2)[X(j(ω-ωc))+X(j(ω+ωc))]

Coherent detection needs carrier phase; envelope detection simplifies the receiver at the cost of carrier power. Single-sideband transmission carries the same information in half the DSB bandwidth.

Frequency modulation places information in instantaneous frequency and trades more bandwidth for improved tolerance to amplitude noise.

PAM carries symbol amplitudes, while QAM uses two quadrature carrier components. Band limitation causes intersymbol interference; raised-cosine and root-raised-cosine pulse shaping balance bandwidth and timing robustness.

OFDM divides data among many orthogonal subcarriers implemented efficiently with the DFT/FFT. A cyclic prefix simplifies multipath equalization, while peak-to-average power ratio, carrier-frequency error, and prefix overhead remain its main costs.

Matched filter

For a known waveform in white noise, the LTI filter that maximizes SNR at the chosen sampling instant is the matched filter; its impulse response is proportional to the time-reversed conjugate of the target waveform. Radar, sonar, digital communications, and correlation-based detection all use this result.

Feedback

Negative feedback gives:

Q=H1/(1+H1H2)

with closed-loop poles satisfying 1+H1H2=0.

Feedback can reduce sensitivity and disturbances, improve tracking, and stabilize an unstable plant; unfavorable phase can also destabilize an otherwise stable system.

Root locus shows pole motion with gain. The Nyquist criterion relates encirclements of -1 to closed-loop stability. Gain and phase margins measure distance to instability; pure delay changes no magnitude but consumes phase margin.

Real processing chains

Separate theory topics merge in real systems:

physical source
-> analog front end
-> anti-alias filter
-> ADC
-> buffer
-> digital filter
-> transform / feature
-> decision / coding
-> transmission / storage

For real-time work, latency, block size, computational cost, memory access, numerical precision, clock drift, and sample loss matter alongside mathematical correctness. Longer FFTs refine frequency spacing but increase block latency; longer FIR filters sharpen transitions but increase computation and group delay.

One framework

time domain          transform domain
------------------   ----------------
convolution          multiplication
derivative/difference frequency weighting
shift                phase factor
exponential input    scalar gain
stability            poles + ROC

Fourier series decomposes periodic signals into harmonics, Fourier transform describes aperiodic spectra, Laplace adds the continuous-time ROC, and the z transform adds the discrete-time ROC.

The continuous-time stability boundary is the jω axis and the discrete-time boundary is the unit circle; sampling connects their frequencies through ω=ΩT.

Key distinctions

• Linearity requires zero input to produce zero output.
• Real-time physical operation requires causality; offline processing may be noncausal.
• BIBO stability requires an absolutely integrable or summable impulse response.
• A discrete sinusoid is periodic only when normalized frequency is rational.
• DTFT has a continuous periodic frequency variable; DFT has finitely many frequency samples.
• FFT is a DFT algorithm, not a new transform.
• Zero padding does not create true frequency resolution.
• Sampling discretizes time; quantization discretizes amplitude.
• The Nyquist rate is a lower bound, not the chosen sampling rate.
• Aliasing is irreversible information loss.
• Decimation requires prior band limitation.
• FIR is finite and nonrecursive; IIR requires pole-stability analysis.
• Linear phase is not zero phase.
• Laplace and z transforms are incomplete without their ROCs.
• Causal stable continuous-time poles lie in the left half-plane; discrete-time poles lie inside the unit circle.
• Finite-precision realizations of the same transfer function can behave differently.
• Open-loop stability alone does not determine closed-loop stability.

References

• Oppenheim, A. V., Willsky, A. S., & Nawab, S. H. Signals and Systems, 2nd ed. Prentice Hall, 1997.
• Oppenheim, A. V., & Schafer, R. W. Discrete-Time Signal Processing, 3rd ed. Pearson, 2010.
• Proakis, J. G., & Manolakis, D. G. Digital Signal Processing: Principles, Algorithms, and Applications, 4th ed. Pearson, 2007.
• Haykin, S., & Van Veen, B. Signals and Systems, 2nd ed. Wiley, 2003.
• Lathi, B. P. Linear Systems and Signals, 2nd ed. Oxford University Press, 2005.
• Lyons, R. G. Understanding Digital Signal Processing, 3rd ed. Pearson, 2011.
• Proakis, J. G., & Salehi, M. Digital Communications, 5th ed. McGraw-Hill, 2007.
• Franklin, G. F., Powell, J. D., & Emami-Naeini, A. Feedback Control of Dynamic Systems, 8th ed. Pearson, 2019.
• Shannon, C. E. “Communication in the Presence of Noise.” Proceedings of the IRE, 37(1), 1949. https://doi.org/10.1109/JRPROC.1949.232969
• Cooley, J. W., & Tukey, J. W. “An Algorithm for the Machine Calculation of Complex Fourier Series.” Mathematics of Computation, 19(90), 1965. https://doi.org/10.1090/S0025-5718-1965-0178586-1
• Welch, P. D. “The Use of Fast Fourier Transform for the Estimation of Power Spectra.” IEEE Transactions on Audio and Electroacoustics, 15(2), 1967. https://doi.org/10.1109/TAU.1967.1161901
• Harris, F. J. “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform.” Proceedings of the IEEE, 66(1), 1978. https://doi.org/10.1109/PROC.1978.10837