EEF206 — Sinyaller & Sistemler Formül Defteri
EEF206 Formül Defteri Sinyaller & Sistemler
Fourier Serisi (CT) Fourier Dönüşümü (CT) AZ Fourier Serisi DTFT / AZFD Z-Dönüşümü Laplace Dönüşümü Örnekleme Teoremi
Sinyaller & Sistemler — Formül Defteri
EEF206 · İTÜ · Fourier, DTFT, Z-Dönüşümü, Laplace, Örnekleme
FS FD / CTFT AZFS / DTFS DTFT / AZFD Z-Dönüşümü Laplace Örnekleme
§ 01
Fourier Serisi (Sürekli Zaman)
Sentez (Synthesis)
Fourier Serisi Gösterimi
x ( t ) = ∑ k = − ∞ ∞ a k e j k ω 0 t x(t) = \sum_{k=-\infty}^{\infty} a_k \, e^{jk\omega_0 t} x ( t ) = k = − ∞ ∑ ∞ a k e j k ω 0 t ω₀ = 2π/T₀ = 2πf₀ (temel açısal frekans)
Analiz (Katsayı Formülü)
Fourier Serisi Katsayıları
a k = 1 T 0 ∫ T 0 x ( t ) e − j k ω 0 t d t a_k = \frac{1}{T_0}\int_{T_0} x(t)\, e^{-jk\omega_0 t}\, dt a k = T 0 1 ∫ T 0 x ( t ) e − j k ω 0 t d t Kutup formu: a k = ∣ a k ∣ e j θ k a_k = |a_k|\,e^{j\theta_k} a k = ∣ a k ∣ e j θ k ∣ a k ∣ = r 2 + w 2 |a_k|=\sqrt{r^2+w^2} ∣ a k ∣ = r 2 + w 2 θ k = arctan ( w / r ) \theta_k=\arctan(w/r) θ k = arctan ( w / r )
Parseval Özelliği
P = 1 T 0 ∫ T 0 ∣ x ( t ) ∣ 2 d t = ∑ k = − ∞ ∞ ∣ a k ∣ 2 P = \frac{1}{T_0}\int_{T_0}|x(t)|^2\,dt = \sum_{k=-\infty}^{\infty}|a_k|^2 P = T 0 1 ∫ T 0 ∣ x ( t ) ∣ 2 d t = k = − ∞ ∑ ∞ ∣ a k ∣ 2 Reel Sinyal Koşulu
x ( t ) ∈ R ⇒ a k = a − k ∗ x(t)\in\mathbb{R} \;\Rightarrow\; a_k = a_{-k}^* x ( t ) ∈ R ⇒ a k = a − k ∗ Yani: ∣ a k ∣ = ∣ a − k ∣ |a_k|=|a_{-k}| ∣ a k ∣ = ∣ a − k ∣ ∠ a k = − ∠ a − k \angle a_k = -\angle a_{-k} ∠ a k = − ∠ a − k
Özellikler
# Özellik x ( t ) x(t) x ( t ) a k a_k a k 1 Doğrusallık A x 1 ( t ) + B x 2 ( t ) Ax_1(t)+Bx_2(t) A x 1 ( t ) + B x 2 ( t ) A a k + B b k Aa_k + Bb_k A a k + B b k 2 Zamanda Öteleme x ( t − t 0 ) x(t-t_0) x ( t − t 0 ) e − j k ω 0 t 0 a k e^{-jk\omega_0 t_0}\,a_k e − j k ω 0 t 0 a k ∣ b k ∣ = ∣ a k ∣ \vert{}b_k\vert{}=\vert{}a_k\vert{} ∣ b k ∣ = ∣ a k ∣ 3 Zaman Tersleme x ( − t ) x(-t) x ( − t ) a − k a_{-k} a − k 4 Konjüge x ∗ ( t ) x^*(t) x ∗ ( t ) a − k ∗ a_{-k}^* a − k ∗ 5 Zamanda Ölçekleme x ( α t ) x(\alpha t) x ( α t ) T 0 / α T_0/\alpha T 0 / α Aynı a k a_k a k α ω 0 \alpha\omega_0 α ω 0 6 Türev d x d t \dfrac{dx}{dt} d t d x j k ω 0 a k jk\omega_0\,a_k j k ω 0 a k 7 İntegral ∫ − ∞ t x ( τ ) d τ \int_{-\infty}^{t}x(\tau)\,d\tau ∫ − ∞ t x ( τ ) d τ a k j k ω 0 \dfrac{a_k}{jk\omega_0} j k ω 0 a k a 0 = 0 a_0=0 a 0 = 0 8 Konvolüsyon x 1 ( t ) ∗ x 2 ( t ) x_1(t)*x_2(t) x 1 ( t ) ∗ x 2 ( t ) T 0 a k b k T_0\,a_k\,b_k T 0 a k b k 9 Çarpım x 1 ( t ) ⋅ x 2 ( t ) x_1(t)\cdot x_2(t) x 1 ( t ) ⋅ x 2 ( t ) ∑ ℓ a ℓ b k − ℓ \sum_\ell a_\ell\, b_{k-\ell} ∑ ℓ a ℓ b k − ℓ
Önemli Çiftler
x ( t ) x(t) x ( t ) T 0 T_0 T 0 a k a_k a k Not e j n ω 0 t e^{jn\omega_0 t} e j n ω 0 t δ [ k − n ] \delta[k-n] δ [ k − n ] tek harmonik cos ( n ω 0 t ) \cos(n\omega_0 t) cos ( n ω 0 t ) 1 2 \tfrac{1}{2} 2 1 k = ± n k=\pm n k = ± n 0 0 0 — sin ( n ω 0 t ) \sin(n\omega_0 t) sin ( n ω 0 t ) 1 2 j \tfrac{1}{2j} 2 j 1 k = n k=n k = n − 1 2 j -\tfrac{1}{2j} − 2 j 1 k = − n k=-n k = − n — ∑ n δ ( t − n T 0 ) \sum_{n}\delta(t-nT_0) ∑ n δ ( t − n T 0 ) 1 T 0 \dfrac{1}{T_0} T 0 1 impuls katarı Periyodik dikdörtgen (darbe genişliği 2 T 1 2T_1 2 T 1 2 T 1 T 0 sinc ( k ω 0 T 1 π ) = sin ( k ω 0 T 1 ) k π \dfrac{2T_1}{T_0}\,\text{sinc}\!\left(\dfrac{k\omega_0 T_1}{\pi}\right) = \dfrac{\sin(k\omega_0 T_1)}{k\pi} T 0 2 T 1 sinc ( π k ω 0 T 1 ) = k π sin ( k ω 0 T 1 ) a 0 = 2 T 1 T 0 a_0=\frac{2T_1}{T_0} a 0 = T 0 2 T 1
sinc(x) = sin(πx)/(πx) · sinc(kT₁/T₀) → k'nın T₁/T₀ oranına bağlı değişim
§ 02
Fourier Dönüşümü (CTFT)
İleri Dönüşüm
Fourier Dönüşümü
X ( j ω ) = ∫ − ∞ ∞ x ( t ) e − j ω t d t X(j\omega) = \int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt X ( j ω ) = ∫ − ∞ ∞ x ( t ) e − j ω t d t Ters Dönüşüm
Ters Fourier Dönüşümü
x ( t ) = 1 2 π ∫ − ∞ ∞ X ( j ω ) e j ω t d ω x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)\,e^{j\omega t}\,d\omega x ( t ) = 2 π 1 ∫ − ∞ ∞ X ( j ω ) e j ω t d ω Özellikler
# Özellik x ( t ) x(t) x ( t ) X ( j ω ) X(j\omega) X ( j ω ) 1 Doğrusallık a x ( t ) + b y ( t ) ax(t)+by(t) a x ( t ) + b y ( t ) a X ( j ω ) + b Y ( j ω ) aX(j\omega)+bY(j\omega) a X ( j ω ) + bY ( j ω ) 2 Zaman Öteleme x ( t − t 0 ) x(t-t_0) x ( t − t 0 ) e − j ω t 0 X ( j ω ) e^{-j\omega t_0}X(j\omega) e − j ω t 0 X ( j ω ) 3 Frekans Öteleme e j ω 0 t x ( t ) e^{j\omega_0 t}x(t) e j ω 0 t x ( t ) X ( j ( ω − ω 0 ) ) X(j(\omega-\omega_0)) X ( j ( ω − ω 0 )) 4 Zaman Tersleme x ( − t ) x(-t) x ( − t ) X ( − j ω ) X(-j\omega) X ( − j ω ) 5 Konjüge x ∗ ( t ) x^*(t) x ∗ ( t ) X ∗ ( − j ω ) X^*(-j\omega) X ∗ ( − j ω ) 6 Ölçekleme x ( a t ) x(at) x ( a t ) 1 ∣ a ∣ X ( j ω a ) \dfrac{1}{\vert{}a\vert{}}X\!\left(j\dfrac{\omega}{a}\right) ∣ a ∣ 1 X ( j a ω ) zaman↓ → frekans↑ 7 Türev (zaman) d n x d t n \dfrac{d^n x}{dt^n} d t n d n x ( j ω ) n X ( j ω ) (j\omega)^n X(j\omega) ( j ω ) n X ( j ω ) 8 İntegral ∫ − ∞ t x ( τ ) d τ \int_{-\infty}^{t}x(\tau)\,d\tau ∫ − ∞ t x ( τ ) d τ 1 j ω X ( j ω ) + π X ( 0 ) δ ( ω ) \dfrac{1}{j\omega}X(j\omega)+\pi X(0)\delta(\omega) j ω 1 X ( j ω ) + π X ( 0 ) δ ( ω ) 9 Konvolüsyon x ( t ) ∗ h ( t ) x(t)*h(t) x ( t ) ∗ h ( t ) X ( j ω ) ⋅ H ( j ω ) X(j\omega)\cdot H(j\omega) X ( j ω ) ⋅ H ( j ω ) 10 Çarpım (Mod.) x ( t ) ⋅ h ( t ) x(t)\cdot h(t) x ( t ) ⋅ h ( t ) 1 2 π X ( j ω ) ∗ H ( j ω ) \dfrac{1}{2\pi}X(j\omega)*H(j\omega) 2 π 1 X ( j ω ) ∗ H ( j ω ) 11 Dualite X ( j t ) X(jt) X ( j t ) 2 π x ( − ω ) 2\pi\,x(-\omega) 2 π x ( − ω ) 12 Parseval ∫ − ∞ ∞ ∣ x ( t ) ∣ 2 d t = 1 2 π ∫ − ∞ ∞ ∣ X ( j ω ) ∣ 2 d ω \displaystyle\int_{-\infty}^{\infty}\vert{}x(t)\vert{}^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}\vert{}X(j\omega)\vert{}^2\,d\omega ∫ − ∞ ∞ ∣ x ( t ) ∣ 2 d t = 2 π 1 ∫ − ∞ ∞ ∣ X ( j ω ) ∣ 2 d ω
Temel Çiftler
x ( t ) x(t) x ( t ) X ( j ω ) X(j\omega) X ( j ω ) Koşul δ ( t ) \delta(t) δ ( t ) 1 1 1 — 1 1 1 2 π δ ( ω ) 2\pi\delta(\omega) 2 π δ ( ω ) — u ( t ) u(t) u ( t ) 1 j ω + π δ ( ω ) \dfrac{1}{j\omega}+\pi\delta(\omega) j ω 1 + π δ ( ω ) — e − a t u ( t ) e^{-at}u(t) e − a t u ( t ) 1 a + j ω \dfrac{1}{a+j\omega} a + j ω 1 a > 0 a>0 a > 0 t e − a t u ( t ) te^{-at}u(t) t e − a t u ( t ) 1 ( a + j ω ) 2 \dfrac{1}{(a+j\omega)^2} ( a + j ω ) 2 1 a > 0 a>0 a > 0 t n e − a t u ( t ) t^n e^{-at}u(t) t n e − a t u ( t ) n ! ( a + j ω ) n + 1 \dfrac{n!}{(a+j\omega)^{n+1}} ( a + j ω ) n + 1 n ! a > 0 a>0 a > 0 e − a ∣ t ∣ e^{-a\vert{}t\vert{}} e − a ∣ t ∣ 2 a a 2 + ω 2 \dfrac{2a}{a^2+\omega^2} a 2 + ω 2 2 a a > 0 a>0 a > 0 e j ω 0 t e^{j\omega_0 t} e j ω 0 t 2 π δ ( ω − ω 0 ) 2\pi\delta(\omega-\omega_0) 2 π δ ( ω − ω 0 ) — cos ( ω 0 t ) \cos(\omega_0 t) cos ( ω 0 t ) π [ δ ( ω − ω 0 ) + δ ( ω + ω 0 ) ] \pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)] π [ δ ( ω − ω 0 ) + δ ( ω + ω 0 )] — sin ( ω 0 t ) \sin(\omega_0 t) sin ( ω 0 t ) π j [ δ ( ω − ω 0 ) − δ ( ω + ω 0 ) ] \dfrac{\pi}{j}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)] j π [ δ ( ω − ω 0 ) − δ ( ω + ω 0 )] — rect ( t 2 T 1 ) \text{rect}\!\left(\dfrac{t}{2T_1}\right) rect ( 2 T 1 t ) 2 sin ( ω T 1 ) ω \dfrac{2\sin(\omega T_1)}{\omega} ω 2 sin ( ω T 1 ) — sin ( W t ) π t \dfrac{\sin(Wt)}{\pi t} π t sin ( W t ) rect ( ω 2 W ) \text{rect}\!\left(\dfrac{\omega}{2W}\right) rect ( 2 W ω ) — ∑ n = − ∞ ∞ δ ( t − n T ) \sum_{n=-\infty}^{\infty}\delta(t-nT) ∑ n = − ∞ ∞ δ ( t − n T ) 2 π T ∑ k δ ( ω − k 2 π T ) \dfrac{2\pi}{T}\sum_k\delta\!\left(\omega-k\dfrac{2\pi}{T}\right) T 2 π ∑ k δ ( ω − k T 2 π ) —
Periyodik Sinyalin FD
X ( j ω ) = ∑ k = − ∞ ∞ 2 π a k δ ( ω − k ω 0 ) X(j\omega) = \sum_{k=-\infty}^{\infty} 2\pi\, a_k\, \delta(\omega - k\omega_0) X ( j ω ) = k = − ∞ ∑ ∞ 2 π a k δ ( ω − k ω 0 ) FS katsayılarından direkt FD elde edilir
İdeal Alçak Geçiren Filtre
H ( j ω ) = { 1 ∣ ω ∣ ≤ ω c 0 ∣ ω ∣ > ω c H(j\omega) = \begin{cases}1 & |\omega|\leq\omega_c \\ 0 & |\omega|>\omega_c\end{cases} H ( j ω ) = { 1 0 ∣ ω ∣ ≤ ω c ∣ ω ∣ > ω c h(t) = sin(ωc·t) / (π·t) → nedensel değil (t<0 için h≠0)
§ 03
Ayrık Zamanlı Fourier Serisi (AZFS)
Sentez
x [ n ] = ∑ k = ⟨ N ⟩ a k e j k ( 2 π / N ) n x[n] = \sum_{k=\langle N\rangle} a_k\, e^{jk(2\pi/N)n} x [ n ] = k = ⟨ N ⟩ ∑ a k e j k ( 2 π / N ) n ω₀ = 2π/N (ayrık temel açısal frekans)N: periyot (tam sayı, rad cinsinden N = 2 π / ω 0 N = 2\pi/\omega_0 N = 2 π / ω 0
Analiz
a k = 1 N ∑ n = ⟨ N ⟩ x [ n ] e − j k ( 2 π / N ) n a_k = \frac{1}{N}\sum_{n=\langle N\rangle} x[n] \, e^{-jk(2\pi/N)n} a k = N 1 n = ⟨ N ⟩ ∑ x [ n ] e − j k ( 2 π / N ) n Periyodik katsayılar: a k + N = a k a_{k+N}=a_k a k + N = a k a k a_k a k
Özellikler
Özellik x [ n ] x[n] x [ n ] a k a_k a k Doğrusallık A x 1 [ n ] + B x 2 [ n ] Ax_1[n]+Bx_2[n] A x 1 [ n ] + B x 2 [ n ] A a k + B b k Aa_k+Bb_k A a k + B b k Zaman Öteleme x [ n − n 0 ] x[n-n_0] x [ n − n 0 ] e − j k ω 0 n 0 a k e^{-jk\omega_0 n_0}\,a_k e − j k ω 0 n 0 a k Frekans Öteleme e j M ( 2 π / N ) n x [ n ] e^{jM(2\pi/N)n}\,x[n] e j M ( 2 π / N ) n x [ n ] a k − M a_{k-M} a k − M Fark Özelliği x [ n ] − x [ n − 1 ] x[n]-x[n-1] x [ n ] − x [ n − 1 ] ( 1 − e − j k ω 0 ) a k (1-e^{-jk\omega_0})\,a_k ( 1 − e − j k ω 0 ) a k Parseval 1 N ∑ n = ⟨ N ⟩ ∣ x [ n ] ∣ 2 = ∑ k = ⟨ N ⟩ ∣ a k ∣ 2 \dfrac{1}{N}\displaystyle\sum_{n=\langle N\rangle}\vert{}x[n]\vert{}^2 = \displaystyle\sum_{k=\langle N\rangle}\vert{}a_k\vert{}^2 N 1 n = ⟨ N ⟩ ∑ ∣ x [ n ] ∣ 2 = k = ⟨ N ⟩ ∑ ∣ a k ∣ 2
ℹ
CT ↔ AZ Karşılaştırma: CT-FS'de sonsuz harmonik (k ∈ Z k\in\mathbb{Z} k ∈ Z k k k e j ω 0 n e^{j\omega_0 n} e j ω 0 n 2 π 2\pi 2 π
§ 04
DTFT — Ayrık Zamanlı Fourier Dönüşümü
İleri Dönüşüm
X ( e j ω ) = ∑ n = − ∞ ∞ x [ n ] e − j ω n X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]\, e^{-j\omega n} X ( e j ω ) = n = − ∞ ∑ ∞ x [ n ] e − j ω n ω: sayısal (dijital) frekans [rad]
Ters Dönüşüm
x [ n ] = 1 2 π ∫ 2 π X ( e j ω ) e j ω n d ω x[n] = \frac{1}{2\pi}\int_{2\pi} X(e^{j\omega})\,e^{j\omega n}\,d\omega x [ n ] = 2 π 1 ∫ 2 π X ( e j ω ) e j ω n d ω İntegral herhangi 2 π 2\pi 2 π [ − π , π ] [-\pi,\pi] [ − π , π ]
⚡ Kritik Özellik — 2π Periyodikliği
X ( e j ( ω + 2 π ) ) = X ( e j ω ) (her zaman!) X\!\left(e^{j(\omega+2\pi)}\right) = X\!\left(e^{j\omega}\right) \quad \text{(her zaman!)} X ( e j ( ω + 2 π ) ) = X ( e j ω ) (her zaman!) → DTFT her zaman dijital frekansta 2π periyodikdir . Frekans eksenini [ − π , π ] [-\pi,\pi] [ − π , π ]
Özellikler
# Özellik x [ n ] x[n] x [ n ] X ( e j ω ) X(e^{j\omega}) X ( e j ω ) 1 Doğrusallık a x [ n ] + b y [ n ] ax[n]+by[n] a x [ n ] + b y [ n ] a X ( e j ω ) + b Y ( e j ω ) aX(e^{j\omega})+bY(e^{j\omega}) a X ( e j ω ) + bY ( e j ω ) 2 Zaman Öteleme x [ n − n 0 ] x[n-n_0] x [ n − n 0 ] e − j ω n 0 X ( e j ω ) e^{-j\omega n_0}X(e^{j\omega}) e − j ω n 0 X ( e j ω ) 3 Frekans Öteleme e j ω 0 n x [ n ] e^{j\omega_0 n}x[n] e j ω 0 n x [ n ] X ( e j ( ω − ω 0 ) ) X(e^{j(\omega-\omega_0)}) X ( e j ( ω − ω 0 ) ) 4 Zaman Tersleme x [ − n ] x[-n] x [ − n ] X ( e − j ω ) X(e^{-j\omega}) X ( e − j ω ) 5 Konjüge x ∗ [ n ] x^*[n] x ∗ [ n ] X ∗ ( e − j ω ) X^*(e^{-j\omega}) X ∗ ( e − j ω ) 6 Konvolüsyon x [ n ] ∗ h [ n ] x[n]*h[n] x [ n ] ∗ h [ n ] X ( e j ω ) ⋅ H ( e j ω ) X(e^{j\omega})\cdot H(e^{j\omega}) X ( e j ω ) ⋅ H ( e j ω ) 7 Çarpım x [ n ] ⋅ h [ n ] x[n]\cdot h[n] x [ n ] ⋅ h [ n ] 1 2 π X ( e j ω ) ∗ H ( e j ω ) \dfrac{1}{2\pi}X(e^{j\omega})*H(e^{j\omega}) 2 π 1 X ( e j ω ) ∗ H ( e j ω ) 8 Fark x [ n ] − x [ n − 1 ] x[n]-x[n-1] x [ n ] − x [ n − 1 ] ( 1 − e − j ω ) X ( e j ω ) (1-e^{-j\omega})X(e^{j\omega}) ( 1 − e − j ω ) X ( e j ω ) 9 Kümülatif Toplam ∑ k ≤ n x [ k ] \sum_{k\leq n}x[k] ∑ k ≤ n x [ k ] 1 1 − e − j ω X ( e j ω ) + π X ( e j 0 ) δ ( ω ) \dfrac{1}{1-e^{-j\omega}}X(e^{j\omega})+\pi X(e^{j0})\delta(\omega) 1 − e − j ω 1 X ( e j ω ) + π X ( e j 0 ) δ ( ω ) 10 Frekans Türevi n ⋅ x [ n ] n\cdot x[n] n ⋅ x [ n ] j d d ω X ( e j ω ) j\dfrac{d}{d\omega}X(e^{j\omega}) j d ω d X ( e j ω ) 11 Parseval ∑ n = − ∞ ∞ ∣ x [ n ] ∣ 2 = 1 2 π ∫ 2 π ∣ X ( e j ω ) ∣ 2 d ω \displaystyle\sum_{n=-\infty}^{\infty}\vert{}x[n]\vert{}^2 = \frac{1}{2\pi}\int_{2\pi}\vert{}X(e^{j\omega})\vert{}^2\,d\omega n = − ∞ ∑ ∞ ∣ x [ n ] ∣ 2 = 2 π 1 ∫ 2 π ∣ X ( e j ω ) ∣ 2 d ω
Temel Çiftler
x [ n ] x[n] x [ n ] X ( e j ω ) X(e^{j\omega}) X ( e j ω ) Koşul δ [ n ] \delta[n] δ [ n ] 1 1 1 — δ [ n − n 0 ] \delta[n-n_0] δ [ n − n 0 ] e − j ω n 0 e^{-j\omega n_0} e − j ω n 0 — u [ n ] u[n] u [ n ] 1 1 − e − j ω + π δ ( ω ) \dfrac{1}{1-e^{-j\omega}}+\pi\delta(\omega) 1 − e − j ω 1 + π δ ( ω ) — α n u [ n ] \alpha^n u[n] α n u [ n ] 1 1 − α e − j ω \dfrac{1}{1-\alpha e^{-j\omega}} 1 − α e − j ω 1 ∣ α ∣ < 1 \vert{}\alpha\vert{}<1 ∣ α ∣ < 1 ( n + 1 ) α n u [ n ] (n+1)\alpha^n u[n] ( n + 1 ) α n u [ n ] 1 ( 1 − α e − j ω ) 2 \dfrac{1}{(1-\alpha e^{-j\omega})^2} ( 1 − α e − j ω ) 2 1 ∣ α ∣ < 1 \vert{}\alpha\vert{}<1 ∣ α ∣ < 1 ∣ α ∣ ∣ n ∣ \vert{}\alpha\vert{}^{\vert{}n\vert{}} ∣ α ∣ ∣ n ∣ 1 − α 2 1 − 2 α cos ω + α 2 \dfrac{1-\alpha^2}{1-2\alpha\cos\omega+\alpha^2} 1 − 2 α cos ω + α 2 1 − α 2 ∣ α ∣ < 1 \vert{}\alpha\vert{}<1 ∣ α ∣ < 1 e j ω 0 n e^{j\omega_0 n} e j ω 0 n 2 π ∑ ℓ δ ( ω − ω 0 − 2 π ℓ ) 2\pi\displaystyle\sum_{\ell}\delta(\omega-\omega_0-2\pi\ell) 2 π ℓ ∑ δ ( ω − ω 0 − 2 π ℓ ) — cos ( ω 0 n ) \cos(\omega_0 n) cos ( ω 0 n ) π ∑ ℓ [ δ ( ω − ω 0 − 2 π ℓ ) + δ ( ω + ω 0 − 2 π ℓ ) ] \pi\sum_\ell[\delta(\omega-\omega_0-2\pi\ell)+\delta(\omega+\omega_0-2\pi\ell)] π ∑ ℓ [ δ ( ω − ω 0 − 2 π ℓ ) + δ ( ω + ω 0 − 2 π ℓ )] — sin ( ω c n ) π n \dfrac{\sin(\omega_c n)}{\pi n} π n sin ( ω c n ) rect ( ω / 2 ω c ) \text{rect}(\omega/2\omega_c) rect ( ω /2 ω c ) ∣ ω ∣ ≤ π \vert{}\omega\vert{}\leq\pi ∣ ω ∣ ≤ π —
Ayrık Zamanlı Filtreler
İdeal AGF (Alçak Geçiren)
H AGF ( e j ω ) = { 1 ∣ ω ∣ ≤ ω c 0 ω c < ∣ ω ∣ ≤ π H_\text{AGF}(e^{j\omega}) = \begin{cases}1 & |\omega|\leq\omega_c \\ 0 & \omega_c<|\omega|\leq\pi\end{cases} H AGF ( e j ω ) = { 1 0 ∣ ω ∣ ≤ ω c ω c < ∣ ω ∣ ≤ π h [ n ] = sin ( ω c n ) π n h[n] = \frac{\sin(\omega_c n)}{\pi n} h [ n ] = π n sin ( ω c n ) İdeal YGF (Yüksek Geçiren)
H YGF ( e j ω ) = 1 − H AGF ( e j ω ) H_\text{YGF}(e^{j\omega}) = 1 - H_\text{AGF}(e^{j\omega}) H YGF ( e j ω ) = 1 − H AGF ( e j ω ) h [ n ] = δ [ n ] − sin ( ω c n ) π n h[n] = \delta[n] - \frac{\sin(\omega_c n)}{\pi n} h [ n ] = δ [ n ] − π n sin ( ω c n ) § 05
Z-Dönüşümü
Tanım
X ( z ) = ∑ n = − ∞ ∞ x [ n ] z − n , z = r e j ω X(z) = \sum_{n=-\infty}^{\infty} x[n]\, z^{-n}, \quad z = re^{j\omega} X ( z ) = n = − ∞ ∑ ∞ x [ n ] z − n , z = r e j ω DTFT ile: X ( e j ω ) = X ( z ) ∣ z = e j ω X(e^{j\omega}) = X(z)\big|_{z=e^{j\omega}} X ( e j ω ) = X ( z ) z = e j ω
Transfer Fonksiyonu
H ( z ) = Y ( z ) X ( z ) = ∑ k = 0 M p k z − k ∑ k = 0 N d k z − k H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M}p_k z^{-k}}{\sum_{k=0}^{N}d_k z^{-k}} H ( z ) = X ( z ) Y ( z ) = ∑ k = 0 N d k z − k ∑ k = 0 M p k z − k Fark denkleminden: ∑ d k y [ n − k ] = ∑ p k x [ n − k ] \sum d_k \text{ y}[n-k] = \sum p_k\text{ x}[n-k] ∑ d k y [ n − k ] = ∑ p k x [ n − k ]
ROC (Yakınsaklık Bölgesi) Kuralları
Sinyal Tipi ROC Şekli Kutup İlişkisi Sağ taraflı (nedensel) ∣ z ∣ > R max \vert{}z\vert{}>R_{\max} ∣ z ∣ > R m a x En büyük kutupun dışı Sol taraflı ∣ z ∣ < R min \vert{}z\vert{}<R_{\min} ∣ z ∣ < R m i n En küçük kutupun içi İki taraflı R 1 < ∣ z ∣ < R 2 R_1<\vert{}z\vert{}<R_2 R 1 < ∣ z ∣ < R 2 Halka bölge Sonlu uzunluklu (FIR) Tüm z z z z = 0 z=0 z = 0 z = ∞ z=\infty z = ∞
Önemli: ROC, kutup içermez. Birim çember ROC içindeyse → DTFT var.
Temel Z-Dönüşümü Çiftleri
x [ n ] x[n] x [ n ] X ( z ) X(z) X ( z ) ROC δ [ n ] \delta[n] δ [ n ] 1 1 1 Tüm z z z δ [ n − n 0 ] \delta[n-n_0] δ [ n − n 0 ] z − n 0 z^{-n_0} z − n 0 z ≠ 0 z\neq 0 z = 0 n 0 > 0 n_0>0 n 0 > 0 u [ n ] u[n] u [ n ] z z − 1 = 1 1 − z − 1 \dfrac{z}{z-1}=\dfrac{1}{1-z^{-1}} z − 1 z = 1 − z − 1 1 ∣ z ∣ > 1 \vert{}z\vert{}>1 ∣ z ∣ > 1 − u [ − n − 1 ] -u[-n-1] − u [ − n − 1 ] 1 1 − z − 1 \dfrac{1}{1-z^{-1}} 1 − z − 1 1 ∣ z ∣ < 1 \vert{}z\vert{}<1 ∣ z ∣ < 1 α n u [ n ] \alpha^n u[n] α n u [ n ] 1 1 − α z − 1 = z z − α \dfrac{1}{1-\alpha z^{-1}}=\dfrac{z}{z-\alpha} 1 − α z − 1 1 = z − α z ∣ z ∣ > ∣ α ∣ \vert{}z\vert{}>\vert{}\alpha\vert{} ∣ z ∣ > ∣ α ∣ − α n u [ − n − 1 ] -\alpha^n u[-n-1] − α n u [ − n − 1 ] 1 1 − α z − 1 \dfrac{1}{1-\alpha z^{-1}} 1 − α z − 1 1 ∣ z ∣ < ∣ α ∣ \vert{}z\vert{}<\vert{}\alpha\vert{} ∣ z ∣ < ∣ α ∣ n α n u [ n ] n\alpha^n u[n] n α n u [ n ] α z − 1 ( 1 − α z − 1 ) 2 \dfrac{\alpha z^{-1}}{(1-\alpha z^{-1})^2} ( 1 − α z − 1 ) 2 α z − 1 ∣ z ∣ > ∣ α ∣ \vert{}z\vert{}>\vert{}\alpha\vert{} ∣ z ∣ > ∣ α ∣ ( n + 1 ) α n u [ n ] (n+1)\alpha^n u[n] ( n + 1 ) α n u [ n ] 1 ( 1 − α z − 1 ) 2 \dfrac{1}{(1-\alpha z^{-1})^2} ( 1 − α z − 1 ) 2 1 ∣ z ∣ > ∣ α ∣ \vert{}z\vert{}>\vert{}\alpha\vert{} ∣ z ∣ > ∣ α ∣ cos ( ω 0 n ) u [ n ] \cos(\omega_0 n)u[n] cos ( ω 0 n ) u [ n ] 1 − cos ( ω 0 ) z − 1 1 − 2 cos ( ω 0 ) z − 1 + z − 2 \dfrac{1-\cos(\omega_0)z^{-1}}{1-2\cos(\omega_0)z^{-1}+z^{-2}} 1 − 2 cos ( ω 0 ) z − 1 + z − 2 1 − cos ( ω 0 ) z − 1 ∣ z ∣ > 1 \vert{}z\vert{}>1 ∣ z ∣ > 1 sin ( ω 0 n ) u [ n ] \sin(\omega_0 n)u[n] sin ( ω 0 n ) u [ n ] sin ( ω 0 ) z − 1 1 − 2 cos ( ω 0 ) z − 1 + z − 2 \dfrac{\sin(\omega_0)z^{-1}}{1-2\cos(\omega_0)z^{-1}+z^{-2}} 1 − 2 cos ( ω 0 ) z − 1 + z − 2 sin ( ω 0 ) z − 1 ∣ z ∣ > 1 \vert{}z\vert{}>1 ∣ z ∣ > 1
Özellikler
Özellik x [ n ] x[n] x [ n ] X ( z ) X(z) X ( z ) ROC Doğrusallık a x 1 + b x 2 ax_1+bx_2 a x 1 + b x 2 a X 1 + b X 2 aX_1+bX_2 a X 1 + b X 2 ⊇ R 1 ∩ R 2 \supseteq R_1\cap R_2 ⊇ R 1 ∩ R 2 Zaman Öteleme x [ n − n 0 ] x[n-n_0] x [ n − n 0 ] z − n 0 X ( z ) z^{-n_0}X(z) z − n 0 X ( z ) R R R Z-Ölçekleme α n x [ n ] \alpha^n x[n] α n x [ n ] X ( z / α ) X(z/\alpha) X ( z / α ) ∣ α ∣ R \vert{}\alpha\vert{}R ∣ α ∣ R Zaman Tersleme x [ − n ] x[-n] x [ − n ] X ( z − 1 ) X(z^{-1}) X ( z − 1 ) 1 / R 1/R 1/ R Z-Domain Türev n ⋅ x [ n ] n\cdot x[n] n ⋅ x [ n ] − z d X d z -z\dfrac{dX}{dz} − z d z d X R R R Konvolüsyon x [ n ] ∗ h [ n ] x[n]*h[n] x [ n ] ∗ h [ n ] X ( z ) H ( z ) X(z)H(z) X ( z ) H ( z ) ⊇ R 1 ∩ R 2 \supseteq R_1\cap R_2 ⊇ R 1 ∩ R 2 Konjüge x ∗ [ n ] x^*[n] x ∗ [ n ] X ∗ ( z ∗ ) X^*(z^*) X ∗ ( z ∗ ) R R R
Ters Z — Kısmi Kesirler (M < N, basit kökler)
H ( z ) = ∑ ℓ = 1 N A ℓ 1 − λ ℓ z − 1 H(z) = \sum_{\ell=1}^{N} \frac{A_\ell}{1-\lambda_\ell z^{-1}} H ( z ) = ℓ = 1 ∑ N 1 − λ ℓ z − 1 A ℓ A ℓ = [ ( 1 − λ ℓ z − 1 ) H ( z ) ] z = λ ℓ A_\ell = \left[(1-\lambda_\ell z^{-1})H(z)\right]_{z=\lambda_\ell} A ℓ = [ ( 1 − λ ℓ z − 1 ) H ( z ) ] z = λ ℓ ROC'a göre sağ/sol taraflı sinyal seçilir:∣ z ∣ > ∣ λ ∣ |z|>|\lambda| ∣ z ∣ > ∣ λ ∣ α n u [ n ] \alpha^n u[n] α n u [ n ] ∣ z ∣ < ∣ λ ∣ |z|<|\lambda| ∣ z ∣ < ∣ λ ∣ − α n u [ − n − 1 ] -\alpha^n u[-n-1] − α n u [ − n − 1 ]
Kararlılık & Nedensellik
Koşul ROC Kutuplar Nedensel ∣ z ∣ > R max \vert{}z\vert{}>R_{\max} ∣ z ∣ > R m a x — BIBO Kararlı Birim çember içinde — Kararlı + Nedensel R max < 1 R_{\max}<1 R m a x < 1 Tüm kutuplar ∣ z ∣ < 1 \vert{}z\vert{}<1 ∣ z ∣ < 1 Kararsız Birim çember ROC dışı ∃ ∣ p k ∣ ≥ 1 \exists\vert{}p_k\vert{}\geq 1 ∃∣ p k ∣ ≥ 1
§ 06
Laplace Dönüşümü
Tanım
X ( s ) = ∫ − ∞ ∞ x ( t ) e − s t d t , s = σ + j ω X(s) = \int_{-\infty}^{\infty} x(t)\,e^{-st}\,dt, \quad s = \sigma + j\omega X ( s ) = ∫ − ∞ ∞ x ( t ) e − s t d t , s = σ + j ω FD ile: X ( j ω ) = X ( s ) ∣ s = j ω X(j\omega)=X(s)\big|_{s=j\omega} X ( j ω ) = X ( s ) s = j ω j ω j\omega j ω
Ters Laplace
x ( t ) = 1 2 π j ∫ c − j ∞ c + j ∞ X ( s ) e s t d s x(t) = \frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty} X(s)\,e^{st}\,ds x ( t ) = 2 π j 1 ∫ c − j ∞ c + j ∞ X ( s ) e s t d s Pratikte kısmi kesirler + tabloya başvur
Temel Laplace Çiftleri
x ( t ) x(t) x ( t ) X ( s ) X(s) X ( s ) ROC δ ( t ) \delta(t) δ ( t ) 1 1 1 Tüm s s s u ( t ) u(t) u ( t ) 1 s \dfrac{1}{s} s 1 σ > 0 \sigma>0 σ > 0 e − a t u ( t ) e^{-at}u(t) e − a t u ( t ) 1 s + a \dfrac{1}{s+a} s + a 1 σ > − a \sigma>-a σ > − a − e − a t u ( − t ) -e^{-at}u(-t) − e − a t u ( − t ) 1 s + a \dfrac{1}{s+a} s + a 1 σ < − a \sigma<-a σ < − a e a t u ( − t ) e^{at}u(-t) e a t u ( − t ) 1 s − a \dfrac{1}{s-a} s − a 1 σ < a \sigma<a σ < a t e − a t u ( t ) te^{-at}u(t) t e − a t u ( t ) 1 ( s + a ) 2 \dfrac{1}{(s+a)^2} ( s + a ) 2 1 σ > − a \sigma>-a σ > − a t n e − a t u ( t ) t^n e^{-at}u(t) t n e − a t u ( t ) n ! ( s + a ) n + 1 \dfrac{n!}{(s+a)^{n+1}} ( s + a ) n + 1 n ! σ > − a \sigma>-a σ > − a e − a t cos ( b t ) u ( t ) e^{-at}\cos(bt)\,u(t) e − a t cos ( b t ) u ( t ) s + a ( s + a ) 2 + b 2 \dfrac{s+a}{(s+a)^2+b^2} ( s + a ) 2 + b 2 s + a σ > − a \sigma>-a σ > − a e − a t sin ( b t ) u ( t ) e^{-at}\sin(bt)\,u(t) e − a t sin ( b t ) u ( t ) b ( s + a ) 2 + b 2 \dfrac{b}{(s+a)^2+b^2} ( s + a ) 2 + b 2 b σ > − a \sigma>-a σ > − a t n t^n t n n ! s n + 1 \dfrac{n!}{s^{n+1}} s n + 1 n ! σ > 0 \sigma>0 σ > 0
Özellikler
Özellik x ( t ) x(t) x ( t ) X ( s ) X(s) X ( s ) ROC Doğrusallık a x 1 + b x 2 ax_1+bx_2 a x 1 + b x 2 a X 1 + b X 2 aX_1+bX_2 a X 1 + b X 2 R 1 ∩ R 2 R_1\cap R_2 R 1 ∩ R 2 Zaman Öteleme x ( t − t 0 ) x(t-t_0) x ( t − t 0 ) e − s t 0 X ( s ) e^{-st_0}X(s) e − s t 0 X ( s ) R R R s s s e s 0 t x ( t ) e^{s_0 t}x(t) e s 0 t x ( t ) X ( s − s 0 ) X(s-s_0) X ( s − s 0 ) R + Re { s 0 } R+\text{Re}\{s_0\} R + Re { s 0 } Ölçekleme x ( a t ) x(at) x ( a t ) 1 ∣ a ∣ X ( s a ) \dfrac{1}{\vert{}a\vert{}}X\!\left(\dfrac{s}{a}\right) ∣ a ∣ 1 X ( a s ) a R aR a R Türev (zaman) d x d t \dfrac{dx}{dt} d t d x s X ( s ) − x ( 0 − ) sX(s)-x(0^-) s X ( s ) − x ( 0 − ) ⊇ R \supseteq R ⊇ R n n n d n x d t n \dfrac{d^n x}{dt^n} d t n d n x s n X ( s ) − s n − 1 x ( 0 − ) − ⋯ − x ( n − 1 ) ( 0 − ) s^n X(s)-s^{n-1}x(0^-)- \cdots - x^{(n-1)}(0^-) s n X ( s ) − s n − 1 x ( 0 − ) − ⋯ − x ( n − 1 ) ( 0 − ) ⊇ R \supseteq R ⊇ R İntegral ∫ − ∞ t x d τ \int_{-\infty}^{t}x\,d\tau ∫ − ∞ t x d τ 1 s X ( s ) \dfrac{1}{s}X(s) s 1 X ( s ) R ∩ { σ > 0 } R\cap\{\sigma>0\} R ∩ { σ > 0 } Konvolüsyon x ( t ) ∗ h ( t ) x(t)*h(t) x ( t ) ∗ h ( t ) X ( s ) H ( s ) X(s)H(s) X ( s ) H ( s ) R 1 ∩ R 2 R_1\cap R_2 R 1 ∩ R 2 s s s − t x ( t ) -t\,x(t) − t x ( t ) d d s X ( s ) \dfrac{d}{ds}X(s) d s d X ( s ) R R R İlk Değer Teor. x ( 0 + ) = lim s → ∞ s X ( s ) x(0^+) = \lim_{s\to\infty} s\,X(s) x ( 0 + ) = lim s → ∞ s X ( s ) Son Değer Teor. lim t → ∞ x ( t ) = lim s → 0 s X ( s ) \lim_{t\to\infty}x(t) = \lim_{s\to 0} s\,X(s) lim t → ∞ x ( t ) = lim s → 0 s X ( s )
Ters Laplace — Kısmi Kesirler
Basit kökler
X ( s ) = ∑ k A k s − p k , A k = ( s − p k ) X ( s ) ∣ s = p k X(s) = \sum_k \frac{A_k}{s-p_k}, \quad A_k = (s-p_k)X(s)\big|_{s=p_k} X ( s ) = k ∑ s − p k A k , A k = ( s − p k ) X ( s ) s = p k m. mertebe tekrarlanan kök p 0 p_0 p 0
A 0 , r = 1 ( m − r ) ! [ d m − r d s m − r ( s − p 0 ) m X ( s ) ] s = p 0 A_{0,r} = \frac{1}{(m-r)!}\left[\frac{d^{m-r}}{ds^{m-r}}(s-p_0)^m X(s)\right]_{s=p_0} A 0 , r = ( m − r )! 1 [ d s m − r d m − r ( s − p 0 ) m X ( s ) ] s = p 0 Diferansiyel denklem → H(s): ∑ a k d k y d t k = ∑ b k d k x d t k \sum a_k \frac{d^k y}{dt^k} = \sum b_k \frac{d^k x}{dt^k} ∑ a k d t k d k y = ∑ b k d t k d k x ⇒ H ( s ) = ∑ b k s k ∑ a k s k \;\Rightarrow\; H(s)=\dfrac{\sum b_k s^k}{\sum a_k s^k} ⇒ H ( s ) = ∑ a k s k ∑ b k s k
Kararlılık & Nedensellik
Koşul ROC / Kutuplar Nedensel σ > σ max \sigma > \sigma_{\max} σ > σ m a x BIBO Kararlı ROC j ω j\omega j ω Kararlı + Nedensel Tüm kutuplar Re { p k } < 0 \text{Re}\{p_k\}<0 Re { p k } < 0 Kararsız ∃ \exists ∃ Re { p k } > 0 \text{Re}\{p_k\}>0 Re { p k } > 0
Frekans Yanıtı
H ( j ω ) = H ( s ) ∣ s = j ω H(j\omega) = H(s)\big|_{s=j\omega} H ( j ω ) = H ( s ) s = j ω Sadece BIBO kararlı sistemlerde geçerlidir
§ 07
Örnekleme Teoremi
İmpuls Treni Örneklemesi
x p ( t ) = x ( t ) ⋅ p ( t ) , p ( t ) = ∑ n = − ∞ ∞ δ ( t − n T ) x_p(t) = x(t)\cdot p(t), \quad p(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT) x p ( t ) = x ( t ) ⋅ p ( t ) , p ( t ) = n = − ∞ ∑ ∞ δ ( t − n T ) X p ( j ω ) = 1 T ∑ k = − ∞ ∞ X ( j ( ω − k ω s ) ) X_p(j\omega) = \frac{1}{T}\sum_{k=-\infty}^{\infty} X\!\left(j(\omega-k\omega_s)\right) X p ( j ω ) = T 1 k = − ∞ ∑ ∞ X ( j ( ω − k ω s ) ) ωs = 2π/T = örnekleme (açısal) frekansıX p X_p X p X X X
Nyquist Örnekleme Kriteri
ω s > 2 ω M ⟺ f s > 2 f M \omega_s > 2\omega_M \quad \Longleftrightarrow \quad f_s > 2f_M ω s > 2 ω M ⟺ f s > 2 f M Terim Tanım ω s = 2 π / T \omega_s = 2\pi/T ω s = 2 π / T Örnekleme frekansı ω M \omega_M ω M Sinyaldeki maksimum frekans 2 ω M 2\omega_M 2 ω M Nyquist oranı (ωs bu değeri aşmalı)ω M \omega_M ω M Nyquist frekansı (yarı Nyquist oranı)
İdeal Yeniden Yapılandırma (AGF ile)
Yeniden Yapılandırma Filtresi
H ( j ω ) = { T ∣ ω ∣ ≤ ω c 0 ∣ ω ∣ > ω c H(j\omega) = \begin{cases}T & |\omega|\leq\omega_c \\ 0 & |\omega|>\omega_c\end{cases} H ( j ω ) = { T 0 ∣ ω ∣ ≤ ω c ∣ ω ∣ > ω c ω M < ω c < ω s − ω M \omega_M < \omega_c < \omega_s - \omega_M ω M < ω c < ω s − ω M Sinc Enterpolasyonu
x r ( t ) = ∑ n x ( n T ) ⋅ ω c T π ⋅ sin ( ω c ( t − n T ) ) ω c ( t − n T ) x_r(t)=\sum_n x(nT)\cdot\frac{\omega_c T}{\pi}\cdot\frac{\sin(\omega_c(t-nT))}{\omega_c(t-nT)} x r ( t ) = n ∑ x ( n T ) ⋅ π ω c T ⋅ ω c ( t − n T ) sin ( ω c ( t − n T )) ω c = ω s / 2 \omega_c=\omega_s/2 ω c = ω s /2 x r ( t ) = ∑ x ( n T ) sinc ( t − n T T ) x_r(t)=\sum x(nT)\text{sinc}\!\left(\frac{t-nT}{T}\right) x r ( t ) = ∑ x ( n T ) sinc ( T t − n T )
Sıfırıncı Dereceden Tutma (ZOH)
ZOH Transfer Fonksiyonu
H 0 ( j ω ) = e − j ω T / 2 ⋅ 2 sin ( ω T / 2 ) ω H_0(j\omega) = e^{-j\omega T/2}\cdot\frac{2\sin(\omega T/2)}{\omega} H 0 ( j ω ) = e − j ω T /2 ⋅ ω 2 sin ( ω T /2 ) ZOH Sonrası Gereken Filtre
H r ( j ω ) = e j ω T / 2 H ( j ω ) 2 sin ( ω T / 2 ) H_r(j\omega) = \frac{e^{j\omega T/2}H(j\omega)}{2\sin(\omega T/2)} H r ( j ω ) = 2 sin ( ω T /2 ) e j ω T /2 H ( j ω ) ZOH → kaba yaklaşım. Doğrusal tutucu (FOH) daha iyi sonuç verir.
Takma (Aliasing)
Takma Koşulu
ω s < 2 ω M ⇒ takma (aliasing)! \omega_s < 2\omega_M \;\Rightarrow\; \text{takma (aliasing)!} ω s < 2 ω M ⇒ takma (aliasing)! Frekans Dönüşümü
ω 0 ⟶ ω s − ω 0 ( ω s / 2 < ω 0 < ω s ) \omega_0 \;\longrightarrow\; \omega_s - \omega_0 \quad (\omega_s/2 < \omega_0 < \omega_s) ω 0 ⟶ ω s − ω 0 ( ω s /2 < ω 0 < ω s ) Sonuç: orijinal frekans daha düşük bir sahte frekansa dönüşür.ωs=ω₀ ise yeniden yapılandırılan sinyal sabittir (DC).
Analog ↔ Dijital Dönüşüm (ADC / DAC)
Analog → Dijital (ADC)
x s ( t ) = x c ( t ) ⋅ ∑ k δ ( t − k T s ) x_s(t) = x_c(t)\cdot\sum_k\delta(t-kT_s) x s ( t ) = x c ( t ) ⋅ k ∑ δ ( t − k T s ) X s ( j ω ) = 1 T s ∑ k X c ( j ( ω − k 2 π T s ) ) X_s(j\omega) = \frac{1}{T_s}\sum_k X_c\!\left(j\!\left(\omega-k\frac{2\pi}{T_s}\right)\right) X s ( j ω ) = T s 1 k ∑ X c ( j ( ω − k T s 2 π ) ) x [ n ] = x c ( n T s ) x[n] = x_c(nT_s) x [ n ] = x c ( n T s ) Dijital → Analog (DAC / İdeal)
h ( t ) = T s ω c π sin ( ω c t ) ω c t h(t) = T_s\frac{\omega_c}{\pi}\frac{\sin(\omega_c t)}{\omega_c t} h ( t ) = T s π ω c ω c t sin ( ω c t ) Koşul: ω s = 2 π / T s > 2 ω c \omega_s = 2\pi/T_s > 2\omega_c ω s = 2 π / T s > 2 ω c
⚡
Özet: ADC: x c ( t ) → × p ( t ) x s ( t ) → d o ¨ n u ¨ s ¸ t u ¨ r u ¨ c u ¨ x [ n ] x_c(t)\xrightarrow{\times p(t)} x_s(t) \xrightarrow{\text{dönüştürücü}} x[n] x c ( t ) × p ( t ) x s ( t ) d o ¨ n u ¨ s ¸ t u ¨ r u ¨ c u ¨ x [ n ] x [ n ] → impuls x s ( t ) → H ( j ω ) x r ( t ) ≈ x c ( t ) x[n]\xrightarrow{\text{impuls}} x_s(t)\xrightarrow{H(j\omega)} x_r(t)\approx x_c(t) x [ n ] impuls x s ( t ) H ( j ω ) x r ( t ) ≈ x c ( t )
Hızlı Referans — Tutucular
Tutucu H ( j ω ) H(j\omega) H ( j ω ) Çıkış Not ZOH (0. dereceden) 2 sin ( ω T / 2 ) ω \dfrac{2\sin(\omega T/2)}{\omega} ω 2 sin ( ω T /2 ) Basamaklı Süreksiz FOH (1. dereceden) 1 T [ sin ( ω T / 2 ) ω / 2 ] 2 \dfrac{1}{T}\left[\dfrac{\sin(\omega T/2)}{\omega/2}\right]^2 T 1 [ ω /2 sin ( ω T /2 ) ] 2 Doğrusal parçalı Sürekli İdeal Sinc T T T Mükemmel Nedensel değil
