An ideal op-amp lead us to an op-amp with input resistance is infinite, so no current flows into either input terminal (the “current rule”) and that the differential input offset voltage is zero (the “voltage rule”).
Ideal operational amplification will have
➝ High input impedance ( R _in = ∞ ) and low output impedance ( R _out = 0 )

➝ R _in = ∞ , so zero input current
➝ Infinite voltage gain
➝ So, it is a voltage controlled voltage source device.

Given data,
The open loop transfer function of a feedback control system is G(s).H(s)=1/(S+1)​ ³
Gain margin=?
→ Here, the formula is -3tan​ ^-1​ (W_{​ pc}​ ) = -180^{​ 0}
tan​ ^-1​ (W_{​ pc}​ ) = -180^{​ 0}​ /3
= -60​ ⁰
W​ _pc​ =√3 rad/sec
G(s).H(s)=1/(S+1)​ ³
|G(s).H(s)|=X= 1/(1+W​ _pc​ ²​ )
=1⁄8
G(s).H(s)=1/X
= 8

(S/N _q)_dB
= (1.76 + 6 n) _dB
(SQNR) ₁ = 1 .76 + 6 n
(SQNR) ₂ = 1 .76 + 6 n(n + 1 ) = 1 .76 + 6 n + 6
(SQNR) ₂ − (SQNR) ₁ = 1 .76 + 6 n + 6 − 1 .76 − 6 n = 6 dB
So for every one bit increase in bits per sample will result is 6 dB improvement in signal to quantization ratio.

The final value theorem is used to find the steady state value of the system output

Let x[n] be causal signal given by x[n] = a​ ⁿ​ u[n]
→ The Z-Transform of x[n] is given by

The ROC is defined by | az ⁻¹ | < 1 or | z| > | a| .
Region of Convergence(RoC)
Region of Convergence for a discrete time signal x[n] is dened as a continuous region in z plane where the Z-Transform converges. In order to determine RoC, it is convenient to represent the Z-Transform as a is rational X(z)=P(z)/Q(z)
1. The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)
2. The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)
3. The RoC of the Z-transform depends on the convergence of the polynomials P(z) and Q(z).

Given, the signal
V (t) = 3 0 sin 100t + 10 cos 300t + 6 sin (500t + π/4)
So, we have
ω ₁ = 100 rads
ω ₂ = 300 rads
ω ₃ = 500 rads
∴ The respective time periods are
T ₁ = 2π/ω₁ = 2π/100 sec
T ₂ = 2π/ω₂ = 2π/300 sec
T ₃ = 2π/ω₃ = 2π/500 sec
So, the fundamental time period of the signal is
LCM (T ₁ , T ₂ , T ₃ ) =LCM (2π,2π,2π)/HCF (100,300,500)
as T ₀ =2π/100
∴ The fundamental frequency, ω₀ =2π/T ₀ = 100 rad/s

→ Shannon Capacity (Noisy Channel) In presence of Gaussian band-limited white noise, Shannon-Hartley theorem gives the maximum data rate capacity.
C=W log​ ₂​ (1+S/N).
where S and N are the signal and noise power, respectively, at the output of the channel.
→ This theorem gives an upper bound of the data rate which can be reliably transmitted over a thermal-noise limited channel.

→ Shannon Capacity (Noisy Channel) In presence of Gaussian band-limited white noise, Shannon-Hartley theorem gives the maximum data rate capacity.
C=B log​ ₂​ (1+S/N).
where S and N are the signal and noise power, respectively, at the output of the channel.
→ This theorem gives an upper bound of the data rate which can be reliably transmitted over a thermal-noise limited channel. =6 * log​ ₂​ (1+16)
=6 * 4.08746284125034
=24.74 kbps

→ Power in sidebands may be calculated as modulation index.
→ Each sideband is 25%. Total modulation index=1.

The open loop transfer function of a feedback control system is G(s).H(s)=1/(S+1)​ ³
Gain margin=?
→ Here, the formula is -3tan​ ^-1​ (W^{​ pc}​ ) = -180​ ⁰ tan​ ^-1​ (W​ ^pc​ ) = -180^{​ 0}​ /3
= -60​ ⁰
W​ ^pc​ =√3 rad/sec
G(s).H(s)=1/(S+1)​ ³
|G(s).H(s)|=X= 1/(1+W​ _pc​ ²​ )
=1⁄8
G(s).H(s)=1/X
= 8

The final value theorem is used to find the steady state value of the system output

Let x[n] be causal signal given by x[n] = a​ ⁿ​ u[n]
→ The Z-Transform of x[n] is given by

Region of Convergence(RoC)
Region of Convergence for a discrete time signal x[n] is dened as a continuous region in z plane where the Z-Transform converges. In order to determine RoC, it is convenient to represent the Z-Transform as a is rational X(z)=P(z)/Q(z)
1. The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)
2. The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)
3. The RoC of the Z-transform depends on the convergence of the polynomials P(z) and Q(z).

Given, the signal
V (t) = 3 0 sin 100t + 10 cos 300t + 6 sin (500t + π/4)
So, we have
ω ₁ = 100 rads
ω ₂ = 300 rads
ω ₃ = 500 rads
∴ The respective time periods are
T ₁ = 2π/ω₁ = 2π/100 sec
T ₂ = 2π/ω₂ = 2π/300 sec
T ₃ = 2π/ω₃ = 2π/500 sec
So, the fundamental time period of the signal is
LCM (T ₁ , T ₂ , T ₃ ) =LCM (2π,2π,2π)/HCF (100,300,500)
as T ₀ =2π/100
∴ The fundamental frequency, ω₀ =2π/T ₀ = 100 rad/s

(S/N _q)_dB
= (1.76 + 6 n) _dB
(SQNR) ₁ = 1 .76 + 6 n
(SQNR) ₂ = 1 .76 + 6 n(n + 1 ) = 1 .76 + 6 n + 6
(SQNR) ₂ − (SQNR) ₁ = 1 .76 + 6 n + 6 − 1 .76 − 6 n = 6 dB
So for every one bit increase in bits per sample will result is 6 dB improvement in signal to quantization ratio.

Power in sidebands may be calculated as modulation index.
→ Each sideband is 25%. Total modulation index=1.

Apply Ohms law, the resistance is equal to the voltage divided by the current. The resulting equivalent resistance is 5/6 Ω.
R=E/I
R=(2*(1/2)V)/3A
= ⅚
Note: This is the cube structure consisting of 12 resistors electrically connected between the 8 vertices. Each resistor is 1 Ω, but any value can be used so long as they are all the same.

In MOSFET fabrication channel length is defined during Poly-Silicon gate patterning process

Shot noise is caused due to random behaviour of charge carriers. Shot noise is generated due to random emission of electrons from cathodes in electron tubes. In semiconductor devices, shot noise is generated due to random generation and recombination of electron-hole pairs