3451 Midterm Flashcards
Impulses
In CT, Dirac delta is an infinite spike at t=0, with an integral of 1.
In DT, the Kronecker delta is 1 when n=0, and 0 otherwise
Sifting/sampling property
Pulse amplitude modulation (PAM)
a ct staircase approximation of x(t) seen as a sum of rectangular pulses each scaled by the sample value
As T approaches 0, staircase converges to the original CT
Bounded variation
typical assumption for real-world is that signals have finite discontinuities and are well-behaved.
Digital processing of analog signals
Typical chain:
anti-aliasing filter: removes high freq before sampling
sample-and-hold (zero-order-hold): produces a staircase signal from CT input
Encoder: Analog to digital converter (ADC): converts sampled values to a digital representation
Decoder (DAC) + zero-order-hold: converts back to a CT output signal if needed
Time shifting
CT: x(t - tnot): shifts signal by t0 in time axis
DT: x[n-n0]: shifts sequence by n0
Complex number z
Cartesian/rectangular form: z = sigmoid + jB,
polar: z = re^j0 where j= root(-1)
Euler’s form: e^j0 = cos + j sin
Frequency variables
X(w), where w=2pif
w - radians/sec, f - hz
for CT, X(w) = integral of x(t)
DT - X(w) = sum of x[n]
Summability (DT)
Integrability (CT)
DT - x[n] is summable if sum of x[n] < infinity
Absolute summability if sum of |x[n]| < inf
Absolute summability -> boundedness and implies x[n] approaches 0 as |n| approaches infinity
Same applies for integrals of CT
but Absolute integrability for CT typically well-behaved but not necessarily bounded
Energy of a signal
CT - E = integral(|x(t)|^2 dt
If E < inf, signal has finite energy
DT is same but sum.
If a signal is absolutely summable/integrable, it has finite energy.
Finite energy does not guarantee absolute integrability
Frequency of CT Pure Sinusoids
pure sinusoid with euler’s - e^jw0t = cost + j sin
if a wheel marks one full rotation in time P(secs) then
w0 = 2pi/P, f = 1/P
sin and cos funcs are special cases of the above, often called pure sinusoids when examining their freq properties
Definition of CT Periodic Signals
ct signal x(t) is periodic if there exists some period P such that
x(t + P) = x(t) for all t
fundamental period p0 is the smallest positive P for which the above holds
corresponding fundamental frequency is w0 = 2pi/ P0
Signal that never repeats is aperiodic - P0 = inf
Combining sinusoids
if x1, and x2 each have periods P1, and P2, then their sum x(t) = ax1 + ax2 is periodic if and only if w1/w2 is a rational number. == P1/P2 is rational
There must be a common factor.
implies -> w0 is a common divisor of w1, and w2
eg. sin)3t) + cos)pi*t) | w1 = 3, w2 = pi. 3/3 is irrational, therefor sum is aperiodic
Finding the Fundamental Period / Frequency
GCD approach, if w1 and w2 share a GCD w0, then the
fundamental frequency w0 leads to the fundamental period p0 = 2pi/w0
Fourier series for CT periodic signals
when x(t) is periodic with fund. period p0, we can write x(t) in a fourier series
cme^…
fourier coefficients cm are found by cm = 1/Po integral 0 to P0, x(t)e^….
Also has a sin + cos form
Special case: Impulse train (periodic lecture)
impulse train can itself be expressed as a fourier series with fundamental period T, so w0 = 2pi/T
Fundamental frequency
Lowest frequency of a periodic waveform
Periodic to aperiodic
Fourier series expresses periodic signals as a potential infinite sum of harmonics of the fundamental freq
Many real-world signals are aperiodic, to handle these extend period to infinity, taking the step from discrete steps to integral over all frequencies (Fourier transform)
Existence of the Fourier Transform
-Bounded var
-Absolute int
-Square-int
-Periodicity
Bounded - function cannot have infinitely many ‘wild’ oscillations in any finite interval
Absolute int - integral(x(t) dt < inf
square ^^
Periodicity - if one of the above holds, a FT typically exists. Real-world signals usually have finite duration and finite amplitude, thus they are absolutelym integrable or atleast square int.
Real signals | Magnitude and Phase
for a general complex signal x(t), its transform X(w) is complex
Where A(w) = |X(w)| is the magnitude spectrum
theta(w) is the phase spectrum, together AwTheta(w) fully characterizes X(w)
Properties of Real x(t)
Even magnitude: A(w) = A(-w)
Odd Phase: theta(w) = -theta(-w)
this means the frequency domain representation of a real signal is symmetric in magnitude and antisymmetric in phase
Phase of a signal
Refers to the position of a wave at a specific point in time within its cycle.
Signifying how much a wave is shifted in time compared to another wave of the same frequency
Comparing Fourier Transform and Fourier Series
Fourier series: Strictly for periodic signals. Outputs discrete set of freqs (m*w0). coeffs cm specify how much of each harmonic is present.
Fourier transforms: For aperiodic (or inf-duration) signals. Is a continuous function of freq w. X(w) gives amp/phase at every w in range of inf.
Since P-> inf for transforms, its more general.
Distribution of Energy in Frequencies (Parseval’s Theorem)
If a signal has finite energy, then parseval’s theorem implies energy measured in time-domain equals energy measured in frequency domain.
Important for analyzing how a signal’s total energy distributes over different frequencies
Frequency Shifting & Modulation
Shifting: a multiplication by e^jw0t in time domain shifts the freq rep by w0.
This property underlines modulation where we shift baseband signal to higher frequencies.
Modulation: Carrier freq wc should be large enough so that these two shifted copies of X(w) do not overlap (otherwise you get interference in freq)
Demodulation: properly chosen wc can recover x(t) from xm(t)
