UP-CONVERSION MIXER

Abstract

According to some embodiments, an up-conversion mixer includes a mixer cell having an output node arranged to provide an output. An input stage is coupled to the mixer cell and arranged to receive an input signal. The mixer cell is configured to generate the output with an up-converted frequency compared to an input frequency of the input signal. The input stage is configured to reduce a third order harmonic term of the output so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum.

Description

TECHNICAL FIELD

The present disclosure relates generally to an integrated circuit and more particularly an up-conversion mixer.

BACKGROUND

An up-conversion mixer is used in many communication applications, e.g., in a single-side band (SSB) communication system. The up-converted signal has an up-conversion term at the local oscillator frequency (f_LO) plus the baseband signal frequency (f_BB), i.e., at (f_LO+f_BB), and a third order harmonic term at (f_LO−3f_BB). A third order distortion suppression P_3d=P(f_LO+f_BB)−P(f_LO−3f_BB), which is the difference of output power between the up-converted signal and the third order harmonic term, is an important factor for a transmitter, e.g., in a Global System for Mobile Communications (GSM). With increasing input power, the third order distortion suppression becomes worse (decreases).

BRIEF DESCRIPTION OF THE DRAWINGS

Reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

FIG. 1 is a plot showing input power vs. output power for an exemplary up-conversion mixer according to some embodiments;

FIG. 2 is a schematic diagram showing an exemplary up-conversion mixer according to some embodiments;

FIG. 3 is a schematic diagram showing an exemplary up-conversion mixer circuit of FIG. 2 according to some embodiments;

FIG. 4 is a plot showing input power vs. output power of the exemplary up-conversion mixer circuit of FIG. 3 according to some embodiments;

FIG. 5 is a schematic diagram showing an exemplary linear transconductance amplifier for an up-conversion mixer according to some embodiments;

FIG. 6 is a schematic diagram showing an exemplary up-conversion mixer circuit with the linear transconductance of amplifier FIG. 5 according to some embodiments;

FIG. 7 is a plot showing input power vs. output power of the exemplary up-conversion mixer circuit of FIG. 6 according to some embodiments; and

FIG. 8 is a flowchart for a method of the exemplary up-conversion mixer circuits in FIG. 3 and/or FIG. 6 according to some embodiments.

DETAILED DESCRIPTION

The making and using of various embodiments are discussed in detail below. It should be appreciated, however, that the present disclosure provides many applicable inventive concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed are merely illustrative of specific ways to make and use, and do not limit the scope of the disclosure.

FIG. 1 is a plot showing input power vs. output power for an exemplary up-conversion mixer according to some embodiments. The up-conversion term at the local oscillator frequency (f_LO) plus the baseband signal frequency (f_BB), i.e., (at f_LO+f_BB), has an output power plot 102 that has a 10 dB/decade slope at lower input power and then becomes relatively flat at higher input power.

The third order harmonic term at (f_LO−3f_BB) of an exemplary conventional up-conversion mixer has an output power plot 104 that has a 30 dB/decade slope at lower input power and then becomes relatively flat at higher input power. The third order distortion suppression (of output signal power) P_3d, i.e., P(f_LO+f_BB)−P(f_LO−3f_BB), has the value 108 at an input power Pin_1. With higher input power, the output power is higher, but P_3d is reduced. With lower input power, the P_3d improves (increases), but the output power is lower, which may require a higher gain power amplifier stage to boost the output power.

The third order harmonic term at (f_LO−3f_BB) of an exemplary up-conversion mixer shows an output power plot 106 having a notch (local minimum, or null close to zero) at the input power Pin_1 on the higher output power side (relatively flat part of the curve 106). Therefore, the P_3d of the exemplary up-conversion mixer improves (increases) by the difference 110 compared to the conventional up-conversion mixer having the P_3d value 108. This allows an improved P_3d and a relatively high output power at the same time for a relatively high power input signal, thus saving the power requirement/consumption of the power amplifier of the output signal. The details of the exemplary up-conversion mixer are described below.

FIG. 2 is a schematic diagram showing an exemplary up-conversion mixer according to some embodiments. An up-conversion mixer circuit 200 has a cascaded transconductance amplifier 202 as an input stage that is coupled to a mixer 204. The mixer 204 multiplies the signal from the cascaded transconductance amplifier 202 with a local oscillator signal, e.g., cos W_LOt to provide an up-converted signal Sig_up. The mixer 204 can be implemented with a Gilbert mixer cell, for example. The cascaded transconductance amplifier 202 includes transconductance amplifiers TA1206 and TA2208 coupled together. TA1206 includes an NMOS transistor M1, and a load transistor R1. TA2208 includes a PMOS transistor M2, and a load transistor R2. In some embodiments, TA1206 may include a PMOS transistor instead of the NMOS transistor M1, and an active device load instead of the resistor R1. In some embodiments TA1208 may include an NMOS transistor instead of the PMOS transistor M2, and an active device load instead of the resistor R2. A voltage V1 is an input voltage signal, Vp is an output voltage signal from TA1206, which is an input voltage signal to TA2208. A voltage Vo is an output voltage signal from TA2208.

Assigning the drain current signal of the NMOS transistor M1 is id₁, and the drain current signal of the NMOS transistor M2 is id₂, id₁ and id₂ are described by the following Equation (1).

id ₁ =gm ₁ v _gs1 +gm ₂ v _gs1 ²+

id ₂ =gm ₂ v _gs2 +gm ₂ v _gs2 ² +gm ₃ v _gs2 ³+

V=v _gs1 ;V _p =v _gs2 =id ₁ *R ₁ ;V _o =id ₂ *R ₂  Equation (1),

where v_gs1 is the source-gate voltage of the NMOS transistor M1, gm_i is the i-th transconductance coefficients of the NMOS transistor M1, i is a positive integer, v_gs2 is the source-gate voltage of the PMOS transistor M2, gm_j′ is the j-th transconductance coefficients of the PMOS transistor M2, j is a positive integer.

The output voltage V_o of the cascaded transconductance amplifier 202 can be described by the following Equation (2):

$\begin{array}{c}{V}_o\sim \left({\mathrm{gm}}_1^{\prime }v_{\mathrm{gs}2}+{\mathrm{gm}}_2^{\prime }v_{\mathrm{gs}2}^2+{\mathrm{gm}}_3^{\prime }v_{\mathrm{gs}2}^3\right)R_2\\ V_p=v_{\mathrm{gs}2}\sim \left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)R_1\\ V_0=\left[{\mathrm{gm}}_1^{\prime }R_1\left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)+{\mathrm{gm}}_2^{\prime }{R_1^2\left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)}^2+{\mathrm{gm}}_3^{\prime }{R_1^3\left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)}^3\right]R_2\\ ={\mathrm{gm}}_1^{\prime }R_1R_2\left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)+{\mathrm{gm}}_2^{\prime }R_1^2{R_2\left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)}^2+{\mathrm{gm}}_3^{\prime }R_1^3{R_2\left({\mathrm{gm}}_1v_{\mathrm{gs}1}+{\mathrm{gm}}_2v_{\mathrm{gs}1}^2\right)}^3\\ =a_1\left(b_1V_i+b_2V_i^2\right)+{a_2\left(b_1V_i+b_2V_i^2\right)}^2+{a_3\left(b_1V_i+b_2V_i^2\right)}^3,\end{array}$ where a₁=gm₁′R₁R₂;a₂=gm₂′R₁²R₂;a₃=gm₂′R₁³R₂

b ₁ =gm ₁ ;b ₂ =gm ₂

For V_i=V_gs1=A·cos w_mt, the cos 3w_mt coefficient is given by

$\begin{array}{cc}\frac{1}{4}\left(2{\mathrm{gm}}_2^{\prime }R_1^2R_2{\mathrm{gm}}_1{\mathrm{gm}}_2+{\mathrm{gm}}_3^{\prime }R_1^3R_2{\mathrm{gm}}_1^3\right)A^3+\frac{15}{16}{\mathrm{gm}}_3^{\prime }R_1^3{\mathrm{gm}}_1{\mathrm{gm}}_2^2A^5.& \mathrm{Equation}\left(3\right)\end{array}$

For a specific v_gs region where gm₃′ is smaller than zero, the cos 3w_mt coefficient can be reduced, e.g., close to zero or minimized, with a circuit parameter design in the up-conversion mixer circuit 200 such that the coefficient A (the amplitude of the input signal Vi) is given by

$\begin{array}{cc}A\sim \sqrt{\left|\frac{8{\mathrm{gm}}_2^{\prime }R_1^2R_2{\mathrm{gm}}_1{\mathrm{gm}}_2+4{\mathrm{gm}}_3^{\prime }R_1^3R_2{\mathrm{gm}}_1^3}{15{\mathrm{gm}}_3^{\prime }R_1^3R_2{\mathrm{gm}}_1{\mathrm{gm}}_2^2}\right|}.& \mathrm{Equation}\left(4\right)\end{array}$

This scheme described above can be referred to as a pre-distortion technique. The cascaded transconductance amplifier 202 reduces the cos 3w_mt term's coefficient resulting from the up-conversion mixer circuit 200, thus the third order harmonic term at (f_LO−3f_BB) is reduced, improving (increasing) the third order distortion suppression P_3d.

FIG. 3 is a schematic diagram showing an exemplary up-conversion mixer circuit of FIG. 2 according to some embodiments. An up-conversion mixer circuit 300 includes input stages 202a and 202b. The input stages 202a and 202b implemented using the cascaded transconductance amplifier 202 in FIG. 2 is coupled to a Gilbert mixer cell 304 that includes NMOS transistors M3, M4, M5, and M6. The input stage 202a includes an NMOS transistor M1a, a PMOS transistor M2a, and load transistors R1a and R2a. The input stage 202b includes an NMOS transistor M1b, a PMOS transistor M2b, and load transistors R1b and R2b. An input signal Vi is coupled across the nodes BBIP and BBIN (gates of NMOS transistors M1a and M1b).

The Gilbert mixer cell 304 is known in the art, and a local oscillator signal is coupled across nodes LOP and LON in the Gilbert mixer cell 304. The nodes LOP and LON are also coupled to a bias voltage VG_LO through a resistor Rg. The up-converted signal output is provided by the nodes RF_P and RF_N. The node RF_P is coupled to the drains of the NMOS transistors M3 and M5 through a coupling capacitor C1. The node RF_N is coupled to the drains of the NMOS transistors M4 and M6 through a coupling capacitor C2.

Because the input stages 202a and 202b (implemented using the cascaded transconductance amplifier 202) reduces the cos 3w_mt term's coefficient resulting from the up-conversion mixer circuit 300, the third order harmonic term at (f_LO−3f_BB) is reduced, improving (increasing) the third order distortion suppression P_3d. Also, the up-conversion mixer circuit 300 has more voltage headroom compared to a conventional up-conversion mixer circuit having four stages from a power supply voltage to ground, since the up-conversion mixer circuit 300 has only two stages from a power supply voltage to ground, e.g., R1a and M1a. Thus, the up-conversion mixer circuit 300 can be implemented for a low voltage application.

FIG. 4 is a plot showing input power vs. output power of the exemplary up-conversion mixer circuit of FIG. 3 according to some embodiments. The third order harmonic term at (f_LO−3f_BB) of the exemplary up-conversion mixer circuit 300 has an output power plot 404 with a notch (local minimum) 406 at the input power −16 dBm. The P_3d of the exemplary up-conversion mixer improves (increases) by about 10 dB, compared to a conventional up-conversion mixer without such a notch 406, for the same up-converted term's output power plot 402. This allows an improved P_3d and a relatively high output power at the same time, saving the power requirement/consumption of the power amplifier for the output signal.

FIG. 5 is a schematic diagram showing an exemplary linear transconductance amplifier for an up-conversion mixer according to some embodiments. The linear transconductance amplifier 500 is a transconductance amplifier that can be used as an input stage to a mixer. The linear transconductance amplifier 500 includes NMOS transistors M7, M8, and M9, a bias voltage BIAS for the gate of the NMOS transistor M9, and a voltage input signal v_gs for the gates of the NMOS transistors M7 and M8. In other embodiments, an input signal can be coupled across the gates of the NMOS transistors M7 and M9 with a built-in bias voltage level. The NMOS transistor M7 operates in a triode (or linear) region. The drains of the NMOS transistors M9 and M8 are coupled to a current source Id.

For a transconductance amplifier with a voltage input signal v_gs and an output current signal id, the output current and transconductance coefficients g_k (the k-th transconductance coefficient, k is a positive integer) are described by the following Equation (5).

$\begin{array}{cc}{i}_d=g_1v_{\mathrm{gs}}+g_2v_{\mathrm{gs}}^2+g_3v_{\mathrm{gs}}^3+\cdots .g_1=\frac{{\mathrm{di}}_d}{{\mathrm{dv}}_{\mathrm{gs}}};g_2=\frac{1}{2}\frac{d^2i_d}{{\mathrm{dv}}_{\mathrm{gs}}^2};g_3=\frac{1}{6}\frac{d^3i_d}{{\mathrm{dv}}_{\mathrm{gs}}^3}& \mathrm{Equation}\left(5\right)\end{array}$

If the first transconductance coefficient g₁ is linear with respect to the input signal v_gs, g₂ is a constant, and g₃ is zero. Accordingly, the third order harmonic term at (f_LO−3f_BB) is reduced (or minimized), e.g., close to zero.

With the drain current signal of the NMOS transistor M9 as id−, and the drain current signal of the NMOS transistor M8 as id+, the g₁ coefficient for the NMOS transistors M7, i.e., g₁₋, is negative, while the g₁ coefficient for the NMOS transistors M8, i.e., g₁₊, is positive. Therefore, the combined g₁ coefficient for the linear transconductance amplifier 500 can be made linear. Because g₁ is linear with respect to v_gs, g₂ is a constant, and g₃ is zero according to Equation (5). Thus, the third order harmonic term at (f_LO−3f_BB) is reduced (or minimized), e.g., close to zero.

More detailed description regarding the linearity of the linear transconductance amplifier 500 can be found in “A 2 GHz 16 dBm IIP3 low noise amplifier in 0.25 μm CMOS technology” (by Y. S. Youn, J. H. Chang, K. J. Koh, Y. J. Lee, and H. K. Yu, in IEEE Int. Solid-State Circuits Conf., San Francisco, Calif., pp. 452-453, February 2003), which is incorporated herein by reference in its entirety. While the exemplary linear transconductance amplifier 500 is implemented using NMOS transistors M7, M8, and M9, a person skilled in the art will appreciate that PMOS transistors can be used in some other embodiments.

FIG. 6 is a schematic diagram showing an exemplary up-conversion mixer circuit with the linear transconductance amplifier of FIG. 5 according to some embodiments. An up-conversion mixer circuit 600 includes input stages 500a and 500b. The input stages 502a and 502b implemented using the linear transconductance amplifier 500 are coupled to the Gilbert mixer cell 304 that includes NMOS transistors M3, M4, M5, and M6. The input stage 500a includes PMOS transistor M7a, M8a, and M9a, and a current source Id1. The input stage 500b includes PMOS transistor M7b, M8b, and M9b, and a current source Id2. An input signal Vi is coupled across the nodes BBIP (gates of PMOS transistors M7a and M8a) and BBIN (gates of PMOS transistors M7b and M8b).

The Gilbert mixer cell 304 is known in the art, and a local oscillator signal is coupled across nodes LOP and LON in the Gilbert mixer cell 304. The nodes LOP and LON are also coupled to a bias voltage VG_LO through a resistor Rg. The up-converted signal output is provided by the nodes RF_P and RF_N. The node RF_P is coupled to the drains of the NMOS transistors M3 and M5. The node RF_N is coupled to the drains of the NMOS transistors M4 and M6.

Because the input stages 500a and 500b (implemented using the linear transconductance amplifier 500) have the g₁ coefficient linear with respect to input signal, the g₃ coefficient is zero, and the third order harmonic term at (f_LO−3f_BB) of the up-conversion mixer circuit 600 is reduced (or minimized), e.g., close to zero. Thus, the third order distortion suppression P_3d is improved (increased). Also, the up-conversion mixer circuit 600 has more voltage headroom compared to a conventional up-conversion mixer circuit having four stages from a power supply to ground, since the up-conversion mixer circuit 600 has only three stages from a power supply voltage to ground, e.g., M7a, M9a, and Id1. Thus, the up-conversion mixer circuit 600 can be implemented for a low voltage application.

FIG. 7 is a plot showing input power vs. output power of the exemplary up-conversion mixer circuit of FIG. 6 according to some embodiments. The third order harmonic term at (f_LO−3f_BB) of the exemplary up-conversion mixer circuit 600 has an output power plot 704 with a notch (local minimum) 706 at the input power −17 dBm. The P_3d of the exemplary up-conversion mixer improves (increases) by about 10 dB, compared to a conventional up-conversion mixer without such a notch 706, for the same up-converted term's output power plot 702. This allows an improved P_3d and a relatively high output power at the same time, saving the power requirement/consumption of the power amplifier for the output signal.

FIG. 8 is a flowchart for a method of the exemplary up-conversion mixer circuits in FIG. 3 and/or FIG. 6 according to some embodiments. At step 802, an input signal is received by an input stage. At step 804, the input stage reduces a third order harmonic term of an output of the up-conversion mixer so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum. At step 806, a mixer cell generates the output with an up-converted frequency compared to an input frequency of the input signal.

In various embodiments, the mixer cell implemented with a Gilbert mixer cell, and a local oscillator signal is supplied to the mixer cell for up-converting the input signal. The input stage reduces a third order harmonic term by cascading two transconductance amplifiers. A first gate of an NMOS transistor is coupled to the input signal. A first drain of the NMOS transistor is coupled to a second gate of a PMOS transistor. A second drain of the PMOS transistor is coupled to the mixer cell.

In various embodiments, the input stage reduces a third order harmonic term by a linear transconductance amplifier in the input stage having a linear first transconductance coefficient with respect to the input signal. The input stage reduces a third order harmonic term by combining a positive first transconductance coefficient and a negative first transconductance coefficient.

According to some embodiments, an up-conversion mixer includes a mixer cell having an output node arranged to provide an output. An input stage is coupled to the mixer cell and arranged to receive an input signal. The mixer cell is configured to generate the output with an up-converted frequency compared to an input frequency of the input signal. The input stage is configured to reduce a third order harmonic term of the output so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum.

According to some embodiments, a method of up-converting an input signal using an up-conversion mixer includes receiving an input signal by an input stage. The input stage reduces a third order harmonic term of an output of the up-conversion mixer so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum. A mixer cell generates the output with an up-converted frequency compared to an input frequency of the input signal.

A skilled person in the art will appreciate that there can be many embodiment variations of this disclosure. Although the embodiments and their features have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the embodiments. Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, and composition of matter, means, methods and steps described in the specification. As one of ordinary skill in the art will readily appreciate from the disclosed embodiments, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed, that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein may be utilized according to the present disclosure.

The above method embodiment shows exemplary steps, but they are not necessarily required to be performed in the order shown. Steps may be added, replaced, changed order, and/or eliminated as appropriate, in accordance with the spirit and scope of embodiment of the disclosure. Embodiments that combine different claims and/or different embodiments are within the scope of the disclosure and will be apparent to those skilled in the art after reviewing this disclosure.

Claims

1. An up-conversion mixer, comprising: a mixer cell having an output node arranged to provide an output; andan input stage coupled to the mixer cell and arranged to receive an input signal,wherein the mixer cell is configured to generate the output with an up-converted frequency compared to an input frequency of the input signal, and the input stage is configured to reduce a third order harmonic term of the output so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum.

2. The up-conversion mixer of claim 1, wherein the input stage comprises a cascaded transconductance amplifier having two transconductance amplifiers coupled together.

3. The up-conversion mixer of claim 2, wherein the cascaded transconductance amplifier comprises an NMOS transistor and a PMOS transistor.

4. The up-conversion mixer of claim 3, wherein the NMOS transistor has a first gate arranged to be coupled to the input signal and a first drain coupled to a second gate of the PMOS transistor.

5. The up-conversion mixer of claim 4, wherein a second drain of the PMOS transistor is coupled to the mixer cell.

6. The up-conversion mixer of claim 1, wherein the mixer cell is a Gilbert mixer cell comprising four NMOS transistors.

7. The up-conversion mixer of claim 1, wherein the input stage comprises a linear transconductance amplifier having a linear first transconductance coefficient with respect to the input signal.

8. The up-conversion mixer of claim 7, wherein the linear transconductance amplifier comprises a first NMOS transistor, a second NMOS transistor, and a third NMOS transistor, a first gate of the first NMOS transistor is coupled to a second gate of the second NMOS transistor, a first drain of the first NMOS transistor coupled to a first source of the third NMOS transistor.

9. The up-conversion mixer of claim 7, wherein the linear transconductance amplifier comprises a first PMOS transistor, a second PMOS transistor, and a third PMOS transistor, a first gate of the first PMOS transistor is coupled to a second gate of the second PMOS transistor, a first drain of the first PMOS transistor coupled to a first source of the third PMOS transistor.

10. The up-conversion mixer of claim 1, further comprising a local oscillator arranged to supply a local oscillator signal to the mixer cell for up-converting the input signal.

11. A method of up-converting an input signal using an up-conversion mixer, comprising: receiving an input signal by an input stage;the input stage reducing a third order harmonic term of an output of the up-conversion mixer so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum; anda mixer cell generating the output with an up-converted frequency compared to an input frequency of the input signal.

12. The method of claim 11, wherein the input stage reducing a third order harmonic term comprises cascading two transconductance amplifiers.

13. The method of claim 12, wherein cascading two transconductance amplifiers comprises: coupling a first gate of an NMOS transistor to the input signal;coupling a first drain of the NMOS transistor to a second gate of a PMOS transistor; andcoupling a second drain of the PMOS transistor to the mixer cell.

14. The method of claim 11, further comprising supplying a local oscillator signal to the mixer cell for up-converting the input signal.

15. The method of claim 11, wherein the mixer cell generating the output is performed utilizing a Gilbert mixer cell coupled to the input stage.

16. The method of claim 11, wherein the input stage reducing a third order harmonic term comprises utilizing a linear transconductance amplifier in the input stage that has a linear first transconductance coefficient with respect to the input signal.

17. The method of claim 11, wherein the input stage reducing a third order harmonic term comprises combining a positive first transconductance coefficient and a negative first transconductance coefficient.

18. An up-conversion mixer, comprising: a Gilbert mixer cell having an output node arranged to provide an output;an input stage coupled to the mixer cell and arranged to receive an input signal; anda local oscillator arranged to supply a local oscillator signal to the Gilbert mixer cell for up-converting the input signal,wherein the Gilbert mixer cell includes four NMOS transistors, the Gilbert mixer cell is configured to generate the output with an up-converted frequency compared to an input frequency of the input signal, and the input stage is configured to reduce a third order harmonic term of the output so that an output power plot of the third order harmonic term with respect to an input power has a notch with a local minimum.

19. The up-conversion mixer of claim 18, wherein the input stage comprises a cascaded transconductance amplifier having two transconductance amplifiers coupled together.

20. The up-conversion mixer of claim 18, wherein the input stage comprises a linear transconductance amplifier having a linear first transconductance coefficient with respect to the input signal.